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TREATISE 


ON 


ALGEBRA, 


CONTAINING 


THE   LATEST  IMPROVEMENTS. 


ADAPTED   TO    THE    USE    OF   SCHOOLS   AND    COLLEGES. 


BY 


CHARLES  W.  HACKLEY,  S.T.D., 

1'KOrESSOP    OK    MATHEMATICS   AND   ASTRONOMY    IN    COLUMniA    COLLEGE,    NEW    YOKK. 


NEW    YORK: 

HARPER   &    BROTHERS,    PUBLISHERS. 

329    &    331    PEARL    STREET, 

i 

FRANKLIN   SQUA  It  E. 

18  6  4. 


i 


Entered,  according  to  Act  of  Congress,  in  the  year  1848  r. 

Harper  <fe  Brothers, 

n  the  Clerk's  Office  of  the  Southern  District  of  New  Yor> 


PREFACE. 


In  the  preparation  of  the  folio  wing  work  no  pains  have  been  spaied 
to  obtair.  from  the  best  sources,  such  as  the  later  treatise  Ip  >r 

repute,  memoirs  of  scientific  bodies,  and  mathematical  journals  in 
English,  French,  and  German,  the  materials  for  a  book  suited  to  the 
present  state  of  mathematical  science  and  the  wants  of  teachers  and 
students. 

The  work  contains  much  that  has  never  before  appeared  in  an  Eng 
lish  dress,  and  almost  every  part  will  be  found  to  present  some  new 
feature.  No  attempt,  however,  has  been  made  at  originality,  unless 
for  the  benefit  of  the  student,  and  in  the  belief  that  the  existing  exposi- 
tions or  processes  were  inferior.  The  object  has  simply  been,  by  any 
and  all  means,  to  make  the  best  book,  without  aiming  so  much  at  indi- 
vidual reputation  as  at  the  author's  own  convenience  and  that  of  other.;, 
devoted,  like  himself,  to  the  noble  task  of  guiding  the  youthful  votaries 
of  science. 

The  French  treatises  furnish  excellent  models  of  the  theory  of 
gebra,  the  German  of  ingenuity  and  brevity  of  notation  and  exposi- 
tion, the  English  of  practical  adaptation  and  variety  of  illustration  and 
example;  and  from  these,  after  a  careful  comparison  of  many  authors 
in  each  language,  demonstrations  have  been  selected  and  introdu 
verbatim  when  they  seemed  incapable  of  improvement ;  but  when- 
ever the  slightest  alteration  or  amalgamation,  or  the  entire  remodeling 
of  them,  could  give  additional  clearness  or  elegance,  the  limas  labor 
has  not  been  spared. 

The  work  will  be  found  to  contain  all  that  is  important  in  the  higher 
parts  of  Algebra,  upon  which  usually  separate  treatises  are  thought 
necessary,  as  well  as  the  elementary  expositions  suited  to  beginners. 
Every  variety  of  symbol  and  of  example  has  been  introduced. 

On  page  XL  those  articles  of  this  volume  are  indicated  which  con- 
stitute a  minimum  coivrse  of  Algebra  requisite  for  the  prosecution  of 
the  higher  branches  of  mathematics.  A  more  extended  course,  such 
as  would  ordinarily  be  advisable,  is  also  pointed  out.  The  rest  may 
very  well  be  reserved  for  reference,  as  the  student's  own  discovery  of 


o  tj  ^>  *<-»  <*  J 


IV  PREFACE. 

his  wants,  in  the  advanced  stages  of  mathematical  pursuit,  shall  call  it 
in  requisition. 

The  author  desires  to  acknowledge  the  effective  assistance  which  lie 
nas  received,  in  revising  the  work  and  superintending  it  through  the 
press,  from  Mr.  J.  J.  Elmendorf,  to  whom  it  is  indebted  for  many  va1 
uable  suggestions. 


CONTENTS. 


Introdcction .  13 

Definitions  and  Notation .                 .  1 

ALGEBRAIC  CALCULUS. 

Reduction  of  Terms .  7 

Addition 8 

Subtraction 11 

Multiplication 13 

Multiplication  by  detached  Coefficients 19 

Division 20 

Division  by  detached  Coefficients 31 

Synthetic  Division 32 

Greatest  common  Measure .  34 

Least  common  Multiple .  40 

ALGEBRAIC   FRACTIONS. 

Reduction 43 

Addition .  47 

Subtraction .  47 

Multiplication 49 

Division ...  4° 


POWERS  AND   ROOTS. 

Powers  and  Roots  of  Monomials 

Addition  and  Subtraction  of  Radicals 

Multiplication  and  Division 

Powers  and  Roots  of  Radicals   . 

Fractional  and  negative  Exponents 

Square  of  Polynomials 

Square  Root  of  Polynomials 

Cube  Root  of  Polynomials 

Square  Root  of  Numbers   . 

A  Square  Root  of  a  whole  Number  can  not  be  a  Fraction 

Property  of  prime  Numbers 

Square  Root  of  whole  Numbers 

Square  Root  by  Approximation 

S(  1 1 tare  Root  of  Fractions    . 

Square  Root  of  decimal  Fractions 

Cube  Root  of  Numbers 

Fourth  Root  of  Numbers 

Rationalizing  binomial  Surds 

Square  Root  of  binomial  Surds 

Binomial  Theorem 

Higher  Roots  of  Numbers  . 

Polynomial  Theorem  . 

Higher  Roots  of  Polynomials 

Fractional  Powers  of  Binomials 

Degree  of  Approximation  of  Series 

Roots  of  imaginary  ExpR-.ssions 


51 

59 

60 

61 

65 

74 

75 

81 

82 

92 

82 

84 

87 

88 

89 

90 

95 

97 

98 

100 

107 

108 

109 

111 

115 

117 


VI  CONTENDS. 

RATIOS   AND    PROPORTION.  Paj. 

Definitions  and  general  Properties •  119 

Propositions  in  Proportion 123 

Examples  in  Proportion 128 


EQUATIONS. 
Preliminary  Remarks 129 

SIMPLE   EQUATIONS. 

Simple  Equations  containing'  one  unknown  Quantity 131 

Examples  in  simple  Equations 134 

Cases  of  Impossibility  and  Indetermination  in  simple  Equations  containing  one  unknown 

Quantity 142 

Bimple  Equations  containing  two  or  more  unknown  Quantities 143 

Examples 144 

General  Formulas  of  Elimination 155 

Problems  producing  simple  Equations 158 

Negative,  indeterminate,  and  infinite  Solutions 173 

Discussion  of  Formulas  furnished  by  general  Equations  of  the  first  Degree,  with  two 

or  more  unknown  Quantities 178 

Problem  of  the  Couriers 181 

Additional  Problems  in  simple  Equations  .  .  183 

Indeterminate  Analysis  of  the  First  Degree 186 

Problems  in  indeterminate  Analysis  .         ...  191 

QUADRATIC   EQUATIONS. 

Definitions,  Divisions •  199 

Pure  Quadratics  containing  one  unknown  Quantity 200 

Examples  in  pure  Quadratics £01 

Pure  Equations  of  higher  Degree 202 

Examples  of  pure  Equations 203 

Complete  Quadratics  containing  one  unknown  Quantity 204 

Examples  in  complete  Quadratics 205 

Solution  of  Quadratics  by  completing  the  Square 208 

Examples 209 

Quadratics  containing  two  unknown  Quantities 218 

Examples 219 

Problems  producing  pure  Equations 224 

Problems  producing  complete  Quadratics -25 

General  Discussion  of  the  Equation  of  the  second  Degree 228 

Problem  of  the  Lights 232 

Problems  solved  by  Quadratics  involving  two  or  more  unknown  Quantities  .         .  234 

Decomposition  of  Trinomials  of  the  second  Degree  into  Factors  of  the  first  Degree       .  239 
Indeterminate  Analysis  of  the  second  Degree  ....  240 

Maxima  and  Minima  ....  2-12 

The  Modulus  of  imaginary  Quantities  242 

Method  of  Mourey  for  avoiding  imaginary  Quantities 244 

PERMUTATIONS   AND    COMBINATIONS. 

Definitions 246 

General  Formulas        .        .      n -  !■? 

Examples 248 

Variations  of  the  general  Formulas .  24!) 

Different  S]  of  Notation 250 

Restricted  Permutations  and  Combinations  of  Numbers .25) 

Calculus  of  Probabilities ...  259 


CONTENTS.  Vll 

METHOD  OF   UNDETERMINED   COEFFICIENTS.  Fag» 

General  Theorem 255 

Decomposition  of  Fractious 256 

Examples    .  257 

LOGARITHMS. 

Definitions  and  Calculation  of  Tables 258 

Geueral  Properties  of  Logarithms      ...  261 

Description  and  Use  of  Tables  ....  262 

Examples  of  the  Application  of  Logarithms  264 

Arithmetical  Complement 264 

Exercises  in  Logarithms 2CG 

Gauss's  Logarithms  for  Sums  and  Differences 268 

Examples 268 

Effect  of  different  Values  of  the  Baso 26d 

Solution  of  exponential  Equations  by  Logarithms 269 

Theorems  in  Logarithms 270 

Exponential  Theorem S72 

Series  for  computing  Logarithms 273 

Method  of  calculating  Napierian  and  common  Logarithms 274 

Examples 275 

PROGRESSIONS. 

Arithmetical  Progression 27b 

Examples 277 

Ten  Formulas  in  Arithmetical  Progression       .        .  ...  .  278 

Geometrical  Progression ,  278 

Examples 280 

Account  of  the  Origin  of  Logarithms  from  Progressions 282 

Ten  Formulas  in  Geometric  Progression 284 

Harmonical  Progression 284 

INTEREST    AND    ANNUITIES. 

Simple  Interest 285 

Present  Value  and  Discount  at  Simple  Interest 286 

Annuities  at  Simple  Interest 287 

Compound  Interest 288 

Present  Value  and  Discount  at  Compound  Interest 291 

Annuities  at  Compound  Interest 291 

Reversion  of  Annuities 292 

Purchase  of  Estates 292 

Reversion  of  Perpetuities 293 

Examples  for  Practice 293 

INTERPOLATION. 

Method  of  first  Differences 294 

Method  of  second  and  higher  Orders  of  Differences 295 

Derivation  of  Formula  for  higher  Orders  of  Differences  by  ihe  Method  of  undetermined 

Coefficients 296 

Example  of  Application  to  Tables     ....  298 

INEQUATIONS. 

Theorems .  298 

Examples  in  Inequations ■  800 


'Ill  CONTENTS. 


GENERAL    THEORY  OF   EQUATIONS. 

NATURE  AND  COMPOSITION  OF  EQUATIONS.  Page 

Definitions   ...  302 

\?f(x)  be  divided  by  x — a,  tbe  Remainder  will  be  /(«) 302 

The  first  Member  of  an  Equation  divisible  by  tbe  Difference  between  the  nuknown 

Quantity  and  a  Root 303 

Every  Equation  has  a  Root 303 

An  Equation  containing  one  unknown  Quantity  has  as  many  Roots  as  there  are  Units 

in  its  Degree ...  308 

Relation  between  the  Roots  and  Coefficients  of  an  Equation 309 

Equations  whose  Coefficients  are  whole  Numbers;  that  of  the  highest  Power  being 

Unity,  can  not  have  Fractional  Roots 312 

Changing  the  Signs  of  the  alternate  Terms  changes  the  Signs  of  the  Roots  .         .        .  312 

Surds  and  impossible  Roots  enter  an  Equation  by  Pairs 313 

All  the  Roots  of  an  Equation  must  be  of  the  Form  a-\-b  \/ — 1 314 

The  Roots  of  two  conjugate  Equations  will  be  Conjugates  of  each  other        .        .         .  314 

DEPRESSION   OR  ELEVATION    OF   THE   ROOTS   OF    EQUATIONS. 

Equations  whose  Roots  are  those  of  the  proposed,  increased  or  diminished  by  a  given 
Quantity 315 

Numbers  between  the  Roots  substituted  for  the  unknown  Quantity  give  results  alter- 
nately Positive  and  Negative 319 

Equation  whose  Roots  separate  those  of  the  proposed 320 

Equal  Roots ' 321 

NUMBER  OF   REAL  AND    IMAGINARY    ROOTS   IN  AN  EQUATION. 

Theorem  of  Sturm 322 

Examples 328 

Horner's  Method  of  resolving  numerical  Equations  of  all  Orders 332 

Examples 333 

Conditions  of  Reality  of  Roots  from  Sturm's  Theorem 340 

Rule  of  Des  Cartes 341 

Theorem  of  Rolle        .     ' 342 

Fourier's  Method  of  separating  the  Roots 345 

Examples 349 

TRANSFORMATION   OF   EQUATIONS. 

To  transform  an  Equation  into  another  whose  second  Term  shall  be  removed  .  .  351 
To  transform  an  Equation  into  another  whose  Roots  shall  be  the  Reciprocals  of  those 

of  the  proposed 352 

To  transform  an  Equation  into  another  whose  Roots  shall  be  any  Multiple  or  Submul- 

tiple  of  those  of  the  proposed 35. 

To  transform  an  Equation  into  another  whose  Roots  shall  be  the  Square  of  those  of  the 

proposed 355 

To  transform  an  Equation  into  another  wanting  any  given  Term 356 

To  transform  an  Equation  into  another  whose  Roots  are  the  Squares  of  the  Differences 

of  those  of  the  proposed 357 

Budan's  Criterion        ....  3.">9 

Degua's  Criterion 361 

LIMITS   OF    THE   ROOTS   OF    EQUATIONS. 

Superior  and  inferior  Limits  of  the  Roots 362 

Newton's  Method  of  finding  the  Limits .  .         .  305 

Wariug's  or  Lagrange's  Method  of  separating  the  Roots        .  .        .  .366 

APPROXIMATION   TO   THE   ROOTS. 

Newton's  Method 3C9 

Method  of  Lagrange  by  continued  Fractions    .        .        .  r;~a 


CONTENTS. 


IX 


BINOMIAL    EQUATIONS. 

When  the  Exponent  is  a  composite  Number   . 
Solution  of  particular  Cases 
Preparatory  Propositions    .... 
Trigonometrical  Solutions  .... 
Multiple  Value  of  Radicals 


375 
376 

378 
37S 
383 


/ 


DETERMINATION   OF   THE  IMAGINARY   ROOTS  OF  EQUATIONS. 

Limits  of  Moduli  of  imaginary  Roots 334 

Lagrange's  Method  of  determining  imaginary  Roots  by  Elimination      ....  385 
Example 38S 

Theory  of  vanishing  Fractions  .        .  39C 

ELIMINATION. 

Resolution  of  Eqaat'.ons  containing  two  or  more  unl.^own  Quantities  of  any  Degree 

whatever 392 

SimpliGcation 394 

Method  of  Labatie 397 

Euler's  Method 404 

Degree  of  the  final  Equation 403 

EXPONENTIAL   EQUATIONS. 

Solution  by  continued  Fractions 407 

By  Logarithms  or  Double  Position 407 

Examples  .  .        •  408 

DEMONSTRATION    OF    BINOMIAL   THEOREM   FOR  ALL   CASES. 

When  the  Exponent  is  a  whole  Number 408 

When  a  Fraction 408 

When  Negative  either  entire  or  fractional 409 


SERIES. 

RECURRING   SERIE8. 

Generation  of  recurring  Series 410 

Return  from  recurring  Series. to  generating  Fraction 411 

To  determine  whether  a  Scries  be  recurring 412 

To  find  the  general  Term  of  a  recurring  Series 414 

Summation  of  Series   '. 415 

Difference  Series 416 

To  separate  the  Roots  of  an  Equation  by  Means  of  difference  Seriea     ....  417 

The  differential  Method  of  summing  Series 419 

Powers  of  the  Temis  of  Progressions         ...  420 

Piling  of  Balls  and  Shells 422 

Variation 425 


SYMMETRICAL   FUNCTIONS. 

Definitions  ....  427 

To  find  the  Sums  of  the  like  and  entire  Powers  of  the  Roots  of  an  Equation         .        .  428 

To  determine  Double,  Triple,  &c,  Functions 430 

Every  rational  and  symmetric  Function  of  the  Root  of  an  Equation  can  be  expressed 

rationally  by  its  Coefficients 431 

Use  of  symmetric  Functions  in  the  transformation  of  Equations 431 

Solution  of  the  Equation  :>f  the  Squares  of  the  Differences '       .  430 

.An  analogous  Method  fo   a  great  number  of  Cases  .        .  ...  433 


A  CONTENTS. 

Page 

Quadratic  Factors  of  Equations         .  •  433 

Elimination  by  Symmetric  Functions        .  •        •  •  436 

GENERAL   SOLUTION  OF  EQUATIONS. 

Method  of  Tschirnhauseu  for  solving  Equations         ...  ....  43a 

Method  of  Lagrange 439 

Examples 441 

GENERAL  EQUATIONS   OF   THE   THIRD   AND  FOURTH   DEGREES. 

Resolution  of  the  E  C[uation  of  the  third  Degree  by  the  Method  of  Cardan      .         .  444 

Irreducible  Case 446 

Solution  of  the  irreducible  Case  by  Trigonometry 41" 

Solution  of  the  reducible  Case  by  Trigonometry 449 

Woolley's  Method  of  resolving  the  Cubic  Equation 449 

Irrational  Expressions  analogous  to  those  obtained  in  the  Resolution  of  Equations  of 

the  third  Degree 452 

Resolution  of  the  Equation  of  the  fourth  Degree 455 


/ 

THE   DIOPHANTINE   ANALYSIS. 

Introductory  Remarks         .        .  457 

Examples .  ......  458 

Questions  fof  Exercise       .  ...  467 

THEORY    OF   NUMBERS. 

Elementary  Propositions ....  468 

The  Forms  and  Relations  of  integral  Numbers,  and  of  their  Sums,  Differences,  and  Prod- 
ucts   469 

Definitions 470 

Divisibility'  of  Numbers 471 

To  find  all  the  Divisors  of  any  Number  whatever 472 

To  form  a  Table  of  prime  Numbers (  472 

To  decompose  a  Number  into  prune  Factors,  and  to  find  afterward  all  its  Divisors        .  473 
To  determine  how  many  Times  a  prime  Number  is  Factor  in  a  Series  of  natural  Num- 
bers   475 

Determination  and  expression  of  perfect  Numbers 476 

To  find  a  Pair  of  amicable  Numbers 476 

Congruous  Numbers  in  general. — Definitions v  477 

Theorems  with  regard  to  congruous  Numbers 477 

No  Algebraic  Formula  can  contain  prime  Numbers  only 479 

Other  Theorems  with  regard  to  prime  Numbers 480 

Primitive  Roots 483 

Theorem  of  Fermat     .  483 

Table  of  primitive  Roots 484 

The  Forms  of  square  Numbers 484 

CONTINUED    FRACTIONS. 

Definitions  .  •  ....  486 

Rule  for  converting  an  irreducible  Fraction  into  a  continued  one 487 

Convergents 488 

Periodic  continued  Fractions 491 

To  develop  any  Quantity  in  a  continued  Fraction   .  493 

Examples 494 

The  Root  of  a  quadratic  Equation  may  be  expressed  in  function  of  the  Coefficients  by 

means  of  continued  Fractions 494 

Resolution  of  the  indeterminate  Equation  of  tho  first  Degree  by  means  of  continued 

Fractions  ...  496 

Method  of  resolving  in  rational  Numbers  indeterminate  Equations  of  the  second  Degree  497 
Gauss's  Mo  hod  of  solving  binomial  Equations  .  501 


A  MINIMUM  COURSE  OF  ALGEBRA. 


Articles*  1-4  inclusive,  6-7,  9-page  17,  Art.  15-p.  26,  Art.  32-46,  48- 
p.  60,  Art.  63-p.  62,  Art.  78-p.  77,  Art.  83-90,  105-110,  119-128,  130-133 
XVI.,  134-143,  p.  138,  139,  Art.  145-p.  150,  Arts.  150,  151,  178-186. 


A  MORE  ENLARGED  COURSE. 


Articles  1-93  inclusive,  101-197,  199-238,  244-258,  298-309,  315-321. 

It  may  be  useful  to  point  out  in  this  connection  a  course  of  mathematica 
study.  1°.  Algebra;  2°.  Geometry  :f  these  two  may  ba  pursued  simultane- 
ously; 3°.  Plane  Trigonometry,  with  its  applications  to  Surveying  and  Navi- 
gation ;  Spherical  Trigonometry,  with  its  applications  to  Practical  and  Nautical 
Astronomy  and  Geodesy  ;J  4°.  Descriptive  Geometry  ;§  5°.  Analytical  Ge- 
ometry ;||  6°.  Theoretic  Astronomy  ;T  7°.  Differential  and  Integral  Calculus 
and  Calculus  of  Variations  ;**  8°.  Mechanics  ;ff  9°.  Optics  ;%t  10°.  Phys- 
ical Astronomy. §§ 

*  The  articles  are  numbered  throughout  the  book  at  the  beginnings  of  paragraphs. 

t  A  treatise  ou  Geometry,  compiled  from  the  latest  and  best  foreign  sources,  has  been 
published  by  the  author. 

t  The  author  has  already  published  a  work  embracing  these  subjects,  a  new  and  greatly 
improved  edition  of  which  will  appear  in  August. 

§  This  branch,  though  it,may  be  omitted  without  destroying  the  connection  between  those 
which  precede  and  follow  it,  is  of  the  highest  advantage  to  the  general  student,  and  invalua- 
ble to  the  engineer.  It  may  be  best  taken  up  in  the  excellent  treatise  by  Professor  Davies. 
In  the  French,  Monge,  the  founder  of  the  science,  has  written  extensively  upon  the  sub 
ject;  there  is  also  a  treatise  by  that  best  of  French  writers  of  elementary  works,  Lefebure 
de  Fourcy.  Professor  DSvies  has  published  a  line  volume  on  the  application  of  descriptive 
geometry  to  shadows  and  perspective. 

||  On  this  subject  there  are  numerous  writers,  Davies,  Pierce,  and  Young,  whose  work  is 
republished  here,  the  author  of  a  treatise  in  the  Library  of  Useful  Knowledge;  and  in  the 
French,  among  the  best,  Biot,  of  whom  there  is  an  English  translation  by  Professor  Smith, 
of  Virginia,  and  Lefebure  de  Fourcy,  whose  work  is  most  generally  preferred.  A  work  on 
this  subject,  by  the  author,  may  be  expected  to  appear  iu  the  course  of  the  next  twelve 
months. 

1T  The  authors  recommended  are  Norton,  Gummery ;  and  in  the  French,  Biot,  of  whom 
there  is  a  translation  in  part,  known  as  the  Cambridge  Astronomy. 

**  This  is  one  of  the  portions  of  mathematical  science  on  which  the  author  proposes  to 
put  forth  a  treatise  at  no  distant  day.  We  have  at  present  on  the  calculus,  Church  and 
Davies,  in  America;  Young,  O'Brien,  and  Walton,  in  England;  Lacroix,  Duhamel,  and 
Moiguo,  who  may  be  mentioned  among  the  numerous  writers  in  France. 

tt  Courtenay's  Boucharlat;  in  French,  Fraueceur  and  Poisson. 

XX  Bache,  Brewster,  Bartlett,  Biot,  and  Jackson.  This  branch  may  be  pursued  to  some 
extent  immediately  after  Geometry. 

§§  The  authors  are  Lagrange  and  Laplace,  of  whose  Mecanique  Celeste  we  have  the 
translation  and  notes  of  Bowditch,  but  for  readers  of  the  French,  the  Systeme  du  Monde 
of  Pontecouland  is  to  be  preferred. 


As  Greek  letters  are  frequently  used  in  the  following  treatise,  lor 
the  convenience  of  those  unaccostunied  to  a  Greek  alphabet,  one  is 
here  inserted.     The  names  of  the  letters  are  given  in  the  last  column 


A 

a 

a 

"kXtpa 

Alpha 

B 

P,6 

b 

B7]ra 

Beta 

r 

y 

g 

Tdfifia 

Gamma 

a 

6 

d 

AeXra 

Delta 

E 

e 

e  short 

vEt/>tA6v 

Epsilon 

Z 

s 

z 

Zf/ra 

Zeta 

H 

n 

e  lonof 

THra 

Eta 

e 

$  e 

th 

Qrjra 

Theta 

i 

i 

i 

'iWTft 

Iota 

K 

K 

k 

Kdmra 

Kappa 

A 

X 

1 

\dfi66a 

Lambda 

M 

v> 

m 

Mil 

Mu 

N 

V 

n 

Nv 

Nu 

m*i 

i 

X 

Zl 

Xi 

0 

0 

o  short 

"0[iiKp6v 

Omicron 

IT 

7T 

P 

m 

Pi 

P 

P 

r 

Tw 

Rho 

2 

o,  ? 

s 

1.iyfia 

Si<mia 

T 

T 

t 

Tai) 

Tau 

T 

V 

u 

TT  xpadv 

Upsilon 

# 

<P 

ph 

$Z 

Phi 

X 

X 

ch 

Xt 

Chi 

* 

V, 

ps 

i'i 

Psi 

3 

(j 

o  long 

'Qfieya 

Omega 

INTRODUCTION. 


In  every  question  of  numbers  there  are  certain  conditions  which  tne 
required  numbers  in  connection  with  the  given  ones  must  fulfill,  whict 
conditions  are  indicated  by  the  question  itself. 

The  solution  has  for  its  object  to  determine  such  required  quantities 
as  will  verify  these  conditions.  It  is  necessary,  therefore,  to  endeavor 
first  to  seize  the  different  relations  by  which  all  the  quantities;  known 
and  unknown,  are  connected  together,  and  to  find  afterward,  by  means 
of  these  relations,  what  opei'ations  ought  to  be  performed  upon  the 
given  quantities  to  obtain  those  which  are  required.  Such  is  the  ob- 
ject proposed  in  that  part  of  mathematics  known  by  the  namo  of  Al- 
gebra. 

To  show  how  the  use  of  letters  and  sisrns  arises,  let  the  following 
simple  problem  be  propose  k 

To  divide  890  dollars  between  three  persons  in  such  a  manner  that  the 
second  shall  have  115  more  than  the  first,  and  the  third  ISO  more  than 
the  second. 

Now  let  us  see  by  what  deductions  the  values  of  the  unknown  num- 
bers may  be  derived. 

Since  the  share  of  the  second  is  115  more  than  that  of  the  first,  and 
the  share  of  the  third  180  more  than  that  of  the  second,  it  will  be  180 
added  to  115,  or  295  more  than  that  of  the  first. 

Then  the  sum  of  the  tln-ee  parts  will  be  formed  of  3  times  the  first 
part,  increased  by  115,  and  also  by  295,  or,  what  is  the  same  thing,  of 
3  times  the  first  part  inci'cased  by  410. 

But  this  is  equal  to  the  sum  to  be  divided,  viz.,  890. 

Then  3  times  the  first  part,  increased  by  410,  is  equal  to  S90. 

Then  3  times  the  first  part  is  equal  to  S90  diminished  by  410,  or  480 

Then  the  first  part  will  equal  the  third  of  480,  or  160  dollars. 

The  first  person,  therefore,  has  160  dollars ;  the  second,  who  must 
nave  115  more,  will  have  275;  and  the  third,  who  was  to  have  ISO 
more  than  the  second,  455  dollai-s.  These  three  sums  united  make 
S90  do'lars,  which  confirms  the  correctness  of  the  result. 

This  example  exhibits  the  kind  of  reasonings  necessary  to  be  em- 
ployed in  the  solution  of  problems  in  numbers;  and  it  will  be  per- 


XiV  INTRODUCTION. 

ceived  that,  to  express  these  reasonings,  it  is  necessary  to  repeat  tie 
quently  a  number  of  words,  designating  the  quantities,  both  known  and 
unknown,  as  xhejirst  part,  the  number  to  be  divided,  &c,  and  other  words 
expressing  the  relations  of  these,  as  increased  by,  diminished  by,  &e. 

To  obviate  the  inconvenience  of  the  periphrases,  by  means  of  which 
the  quantities  which  enter  into  the  question  are  distinguished,  it  is  cus- 
tomary to  represent  these  quantities  by  letters.  Ordinarily,  the  giveu 
quantities  are  represented  by  the  first  letters  of  the  alphabet,  a,b,c  . . . , 
and  the  required  or  unknown  by  the  last,  x,  y,  z  .  .  . 

The  relations  are  expressed  by  signs.  Thus,  increased  by  is  written 
+  ;  diminished  by  is  written  — ;  multiplied  by  is  written  X  ;  or,  a  mul- 

iplied  by  b,  simply  thus,  ab  ;  a  divided  by  b,  thus,  j;  a  equal  to  h, 

vus,  a-=b. 

The  reasoning  of  the  above  example  may,  with  the   aid  of  such 

ibridgments,  if  x  denote  the  first  share,  be  written  briefly  thus : 

x 

x+115 

3+115  +  180 

3x  +  410  =  890 

3z  =  890— 410 

3z  =  480 

480        ' 
x  =— =160 

If  the  numbers  had  been  different  in  the  above  problem,  the  method 
"proceeding  would  have  been  precisely  the  same. 

Thus,  if  1250  had  been  the  sum  to  be  divided,  170  the  excess  of  the 
second  part  over  the  first,  and  220  the  excess  of  the  third  over  the  sec- 
ond, the  reasoning  would  have  had  the  same  form,  as  seen  below. 

x  share  of  the  1st,  230 

x  +  170  170 

.-r+170+220  share  of  the  2d,  400 


3a;+560  =  1250  220 

3x=  1250— 560  share  of  the  3d,  620 

3a;  =  690  Proof. 

690  230 

x  =  —r-  =  230 


400 

620 

1250 

All  these  individual  cases  of  the  same  kind  may  be  generalized,  thus  : 
Let  a  represent  the  number  to  be  divided  ;  b  the  excess  of  die  second 
over  the  first  share  ;  c  that  of  the  third  over  the  second.     The  reason 
mrr  will  then  stand  as  follows  : 


INTRODUCTION.  *» 

» 

X 

z+b 

x+&-f-c 


3x+2b+c=a 

3x—a — 2b — c 

a — 2b — c 


x- 


The  last  expression,  x=. ,  shows  what  operations  ought  to  be 

performed  upon  the  given  numbers  to  produce  the  required,  and  may 
be  interpreted  into  the  following  rule. 

Subtract  double  the  excess  of  the  second  share  over  the  first,  together 
with  the  excess  of  the  third  over  the  second,  from  the  number  to  be  divided, 
and  divide  the  remainder  by  3.  The  result  will  be  the  first  share  re- 
quired. 

Applying  this  rule  to  the  first  case  above,  we  have 

115x2=230  890  and  to  the  2d,  170 

180  2 

'       "         410  340 


3)480  220  1250 

160  Ans.  560 

3)690 
~230  Ans. 

The  expression  x=. ,  from  which  the  rule  to  be  applied  is 

derived,  is  called  a  general  formula,  or  simply  a  formula  from  which, 
instead  of  from  the  rule,  the  answers  in  the  particular  cases  may  be 
obtained  by  substitution  ;  thus, 

in  the  1st  case,  in  the  2d  case, 

890—230—180     4S0     „rtrt  1250—2x170—220     690 

z— =——=160,    x— = =230 

3  3  3  3 

The  nature  and  utility  of  algebra  being  thus  briefly  indicated,  we 
proceed  to  give  in  detail,  first,  the  methods  of  representing  quantities, 
and  all  possible  relations  and  combinations  of  them,  and  afterward  the 
use  of  these  methods  in  the  solution  of  questions,, 


ALGEBRA, 


DEFINITIONS  AND  NOTATION. 

1.  Algebra  is  a  species  of  short-hand  writing  which,  by  the  aid  of  certain 
symbols,  serves  to  abridge  and  generalize  propositions  relating  to  numbers.* 

A  Proposition  is  any  thing  propounded  as  true.  If  it  express  the  proper- 
ties or  relations  of  quantity,  it  is  a  mathematical  proposition.  If  it  be  self- 
evident,  it  is  called  an  axiom.  If  it  require  demonstration,  it  is  called  a  theorem ; 
and  if  it  propose  something  to  be  done,  or  that  some  required  or  unknown 
quantity  be  found,  it  is  called  a  problem. 

Symbols  may  be  divided  into  symbols  of  quantity,  and  symbols  of  relation 
commonly  called  signs. 

2.  The  principal  symbols  employed  in  algebra  are  the  following : 

I.  The  letters  of  the  alphabet,  a,  b,  c,  &c,  which  are  employed  to  denote 
the  numbers  which  are  the  object  of  our  reasonings. 

When  the  Roman  letters  are  exhausted,  or  when  a  marked  distinction  is  de 
sirable  between  the  different  classes  of  quantities  employed,  the  Greek  letters 
are  also  used  as  representatives  of  quantity.  If  different  quantities  of  the  same 
general  nature  are  used  together,  it  is  a  common  custom  to  represent  tliem  by 
the  same  letter,  distinguishing  them  from  one  another  by  accents,  or  small 
numbers  written  below ;  thus,  a,  a',  a",  a'",  aiv,  are  representatives  of  differ 
ent  quantities,  and  are  read  a,  a  prime,  a  second,  &c. ;  and  au  Og,  a3,  &c, 
may  be  read  a  one  subscript,  a  two  subscript,  and  so  on. 

A  similar  effect  is  produced  by  using  large  and  small  letters ;  thus,  the  di- 
ameter of  a  small  circle  being  represented  by  d,  that  of  a  larger  may  be  by  D. 

It  is  customary,  in  some  cases,  to  represent  quantities  by  symbols,  which 
indicate  distinctly  the  nature  of  the  quantities  represented.  Thus,  the  six 
trigonometrical  quantities,  which  are  known  by  the  names  of  sine,  tangent, 
secant,  cosine,  cotangent,  cosecant,  are  represented  by  the  symbols  sin,  tan, 
sec,  cos,  cot,  cosec ;  and  the  astronomical  quantities,  the  longitude  of  the 
sun,  the  longitude  of  the  moon,  and  the  longitude  of  a  node,  are  represented 
by  the  symbols  O ,   })  ,  and  t?  • 

*  In  the  operations  of  Arithmetic,  with  the  exception  of  those  which  relate  to  compound 
numbers,  quantities  are  considered  as  composed  of  units,  but  the  kind  of  unit  is  not  noticed, 
only  the  number.  In  Algebra,  neither  the  kind  nor  number  of  units  of  which  a  quantity 
is  composed  is  regarded,  and  often  the  quantity  is  not  considered  as  composed  of  units  at 
all.  The  idea  of  number  may,  however,  always  be  introduced,  and  it  is  best  to  keep  it  in 
mind  in  the  beginning  of  Algebra.  As  in  Arithmetic  the  rules  of  addition,  multiplication, 
proportion,  &c,  are  the  same,  whatever  be  the  kind  of  units  which  the  numbers  employed 
represent,  so  in  Algebra  these  rules  are  the  same,  whatever  be  either  the  kind  or  num- 
ber of  units  in  the  quantities  employed  (upon  which  the  operations  are  performed).  In 
every  part  of  Algebra,  processes  analogous  to  those  prescribed  by  the  rules  of  Arithmetic 
are  in  use.  Hence,  and  because  of  its  character  of  generalization,  it  was  called  by  New 
ton  General  Arithmetic.  Algebra,  however,  presents  many  relations  of  quantity  of  which 
Arithmetic  takes  uo  cognizance. 

A 


2  ALGEBRA. 

These  are  the  symbols  of  quantity. 
The  following  are  symbols  of  relations  : 

II.  The  sign  -}-,  which  is  named  plus',  and  is  employed  to  denote  the  addi 
tion  of  two  or  more  numbers. 

Thus,  12-J-30  signifies  12  plus  30,  or,  12  augmented  by  30.  In  like  manner, 
a  -f-  b  signifies  a  plus  b,  or,  the  number  designated  by  a  augmented  by  the 
number  designated  by  b. 

III.  The  sign  — ,  which  is  named  minus,  and  is  employed  to  denote  the 
subtraction  of  one  number  from  another. 

Thus,  54 — 23  signifies  54  minus  23,  or,  54  diminished  by  23.  In  like  man- 
ner, a  —  b  signifies  a  minus  b,  or,  the  number  designated  by  a  diminished  by 
the  number  designated  by  b. 

The  sign  ~  is  sometimes  employed  to  denote  the  difference  of  two  num- 
bers, when  it  is  not  known  which  is  the  greater.  Thus,  a~&  signifies  the 
difference  of  a  and  b,  when  it  is  not  known  whether  the  number  designated  by 
a  be  less  or  greater  than  the  number  designated  by  b. 

IV.  The  sign  X  >  which  may  be  read  into,  is  employed  to  denote  the  multi- 
plication of  two  or  more  numbers. 

Thus,  72  X  26  is  read  72  into  26,  or,  72  multiplied  by  26.     In  like  manner, 
a  X  b  signifies  a  into  b,  or,  a  multiplied  by  b  ;  and  a  X  b  X  c  signifies  the  con 
tinued  product  of  the  numbers  designated  by  a,  b,  c  ;  and  so  on  for  any  num- 
ber of  factors. 

The  process  of  multiplication  is  also  frequently  indicated  by  placing  a  point 
between  the  successive  factors ;  thus,  a  .b  .  c  .  d  signifies  the  same  thing  as 
axbxcxd. 

In  general,  however,  when  numbers  are  represented  by  letters,  their  multi- 
plication is  indicated  by  writing  the  letters  in  succession,  without  the  interpo- 
sition of  any  sign.  Thus,  ab  signifies  the  same  thing  as  a  .  b,  or  a  X  b  ;  and 
abed  is  equivalent  to  a  .  b  .  c .  d,  or  a  X  b  X  c  X  d. 

Factors  expressed  by  letters  are  called  literal  factors,  and  those  expressed 
by  numbers  numerical  factors. 

It  must  be  remarked,  that  the  notation  a .  b,  or  ab,  can  be  employed  only 
when  the  numbers  are  designated  by  letters;  if,  for  example,  we  wished  to  rep- 
resent the  product  of  the  numbers  5  and  6  in  this  manner,  5 .  6  would  be  con 
founded  with  an  integer  followed  by  a  decimal  fraction,  and  56  would  signify 
the  number  ffty-six,  according  to  the  common  system  of  notation. 

For  the  sake  of  brevity,  however,  the  multiplication  of  numbers  is  some 

times  expressed  by  placing  a  point  between  them  in  cases  where  no  ambiguity 

can  arise  from  the  use  of  this  symbol.     Thus,  1.2.3.4,  may  represent  the 

,276 
continued  product  of  the  numbers  1,  2,  3,  4 ;  and  -  .  -  .  —  may  represent 

2  7  6 

the  product  of  -x,  -,  and  — . 

V.  The  sign  -4-,  which  is  named  by,  and  whon  placed  between  two  num- 
bers is  employed  to  denote  that  the  former  is  to  be  divided  by  the  latter. 

Thus,  244-6  signifies  24  by  6,  or,  24  divided  by  6.  Id  ike  manner,  a~b 
signifies  a  by  b,  or,  a  divided  by  b. 

Two  dots  without  the  horizontal  line  between  are  also  the  sign  of  division. 
This  form  of  the  sign  is  used  in  proportions,  where  either  of  the  two  quantities 


DEFINITIONS  AND  NOTATION.  3 

between  which  it  is  placed  may  be  regarded  as  the  dividend,  and  the  other  tne 
divisor.     It  is  analogous,  in  this  respect,  to  the  sign  ~  in  subtraction. 

In  general,  however,  the  division  of  two  numbers  is  indicated  by  writing  the 
dividend  above  the  divisor,  and  drawing  a  line  between  them.     Thus,  21-^-G 

j         7  24       i  a 

and  a-^-b  are  usually  written  —  and  r. 

J  b  b 

Every  fraction,  then,  expresses  the  quotient  of  its  numerator,  divided  by  its 
denominator.  Thus,  f  of  a  unit  may  be  regarded  as  composed  of  two  parts  : 
the  one,  the  third  of  one  unit,  and  the  other,  the  third  of  another  unit ;  or 
both  together,  the  third  of  2  units,  or  the  quotient  of  2  divided  by  3.  This 
reasoning  may  be  generalized. 

VI.  The  sign  =,  called  the  sign  of  equality,  and  read  is  equal  to,  when 
placed  between  two  numbers  denotes  that  they  are  equal  to  each  other. 

Thus,  56-\-6=62  signifies  that  the  sum  of  56  and  6  is  equal  to  62.  In  like 
manner,  a  =  £>  signifies  that  a  is  equal  to  b,  and  a-\-b=c — d  signifies  that  a 
plus  b  is  equal  to  c  minus  d,  or  that  the  sum  of  the  numbers  designated  by  a 
and  b  is  equal  to  the  difference  of  the  numbers  designated  by  c  and  d. 

VII.  The  sign  <^,  which  is  read  is  unequal  to,  and  when  placed  between 
two  numbers  denotes  that  one  of  them  is  greater  than  the  other,  the  opening 
of  the  sign  being  turned  toward  the  greater  number. 

Thus,  a>6  signifies  that  a  is  greater  than  b,  and  a<&  signifies  that  a  is 
less  than  b. 

VIII.  The  coefficient  is  a  sign  which  is  employed  to  denote  that  a  number 
designated  by  a  letter,  or  some  combination  of  letters,  is  addnd  to  itself  a  cer- 
tain number  of  times. 

Thus,  instead  of  writing  a+a-j-a-f-a-f-a,  which  represents  5  a's  added 
together,  we  write  5a.  In  like  manner,  10a&  will  signify  the  same  thing  as 
ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab-\-ab,  or  ten  times  the  product  of 
a  and  b. 

The  numbers  5  and  10  here  are  coefficients. 

The  coefficient,  then,  is  a  number,  -written  to  the  left  of  another  number 
represented  by  one  or  more  letters,  and  denotes  the  number  of  times  that  the 
given  letter,  or  combination  of  letters,  is  to  be  repeated. 

Or  the  coefficient  is  the  numerical  factor  written  before  one  or  more  literal 
factors. 

When  no  coefficient  is  expressed,  the  coefficient  1  is  always  understood , 
thus,  la  and  a  signify  the  same  thing. 

In  a  more  enlarged  sense,  one  literal  factor  may  be  regarded  as  the  coeffi- 
cient of  another,  especially  when  the  former  is  one  of  the  first,  and  the  latter 
one  of  the  last  letters  of  the  alphabet.     Thus,  in  the  expression  ax,  a  may  be 
called  the  coefficient  of  x.     So,  also,  in  the  expression  of  abxy,  ab  may  be  re    . 
garded  as  the  coefficient  ofxy. 

IX.  The  exponent,  or  index,  is  a  sign  which  is  employed  to  denote  that  a 
number  designated  by  a  letter  is  multiplied  by  itself  a  certain  number  of  times. 

Thus,  instead  of  writing  ax  aXaXaX  a,  or  aaaaa,  which  represents 
five  a's  multiplied  together,  we  write  a5,  where  5  is  called  the  exponent  or 
index  of  a.  Similarly,  &X&X&X&X&X<,'X&X&X&X&»  or  b  .b  .b. 
b  .  b.  b  .  b  .  b  .  b  .  b,  or  bbbbbbbbbb  ;  or  the  continued  product  of  10  i's  is  written 
more  briefly  b10,  where  10  is  the  exponent  or  index  of  b. 

The  exponent  or  index  of  a  number  is,  therefore,  a  number    *ritten  a  little 


4  ALGEBRA. 

above  a  letter  to  the  right,  and  denotes  the  number  of  times  which  the  number 
designated  by  the  letter  enters  as  a  factor  into  a  product.  When  no  exponent 
is  expressed,  the  exponent  1  is  always  understood ;  thus,  a1  and  a  signify  the 
same  thing. 

The  products  thus  formed  by  the  successive  multiplication  of  the  same 
number  by  itself,  are  in  general  called  the  powers  of  that  number.  Thus,  a  ia 
the  first  power  of  a  ;  aXa=aa=a2  is  (he  second  power  of  a,  or  the  square 
of  a  ;  aaa=a3  is  the  third  power,  or  cube  of  a ;  aaaaa=a5  is  the  fifth  powet 

of  a,  and  aaaa ton  factors  =  an,  is  the  nth  power  of  a,  or  the  powei 

of  a  designated  by  the  number  n. 

X.  The  square  root  of  any  expression  is  that  quantity  which,  when  multi- 
plied by  itself,  will  produce  the  proposed  expression,  and  is  generally  denoted 
by  the  symbol  sf ,  which  is  called  the  radical  sign.  Thus,  the  square  root 
of  9  is  -v/9=3,  and  -\/a°=a,  is  the  square  root  of  a";  for  in  the  former  case 
3x3=9,  and  in  the  latter  aX«=«2« 

XI.  The  cube  root  of  any  expression  is  that  quantity  which,  when  multi- 
Dlied  twice  by  itself,  -will  produce  the  proposed  expression.  The  fourth,  or 
biquadrate  root  of  any  expression  is  that  quantity  which,  when  multiplier' 
three  times  by  itself,  produces  the  given  expression ;  and  the  nth  root  of  any 
expression  is  that  quantity  which,  multiplied  (n — 1)  times  by  itself,  produces 
the  proposed  expression.  Thus,  the  cube  root  of  8  is  2;  for  2x2X2=8, 
the  fourth  root  of  a*  is  a  ;  for  a  .  a  .  a  .  a=a4,  and  the  nth  root  of  xn  is  x  ;  for 
x  X  x  X  x  . . . .  to  n  factors  =x  .x.x  .x ....  ton  factors  =  xn: 

The  roots  of  expressions  are  frequently  designated  by  fractional  or  decimal 
exponents,  the  figure  in  the  numerator  of  the  fractional  exponent  denoting  the 
power  to  which  the  expression  is  to  be  raised  or  involved,  and  the  figure  in 
the  denominator  denoting  the  root  to  be  extracted  or  evolved.  Thus,  the 
symbol  of  operation  for  the  square  root  of  a  is  either  -y/  a  or  a- ;  for  the  cube 
root  it  is  y a,  or  a*;  for  the  fourth  root,  $/a,  or  a*;  and  y/a,  or  a5,  denotes 
the  7i th  root  of  a.     Also,  %/a5,  or  a%  denotes  the  sixth  root  of  the  fifth  power 

m 

of  a  ;  and  aa,  or  Vflm>  signifies  the  wth  root  of  the  with  power  of  a.* 

XII.  A  rational  quantity  is  one  which  can  be  expressed  without  a  radical 
sign  or  fractional  exponent,  as  3mn,  or  bx^y*. 

XIII.  An  irrational  quantity  is  a  root  which  can  not  be  exactly  extracted, 
and  is  expressed  by  means  of  the  radical  sign  -\/,  or  a  fractional  exponent,  as 

V2  yd*,  or  x*yK 

XIV.  The  reciprocal  of  any  quantity  is  unity  divided  by  that  quantity; 

i  •    ,      1     1     1     1     , 

thus,  the  reciprocals  of  a?,  Xs,  y5,  z*,  are  respectively  -3,  — ,  -5,  —^ ;  but  the 

following  notation  is  generally  used,  as  being  more  commodious  :  thus,  the 
fractions  —%,  3  -5,  -j,  are  expressed  by  a~%  x~s,  y~5,  z~l* 

It  will  follow  from  the  above,  and  from  the  rule  for  division  of  fractions,  that 
the  reciprocal  of  a  fraction  is  the  fraction  inverted.     Thus,  the  reciprocal  of 

a  .    1      b 

r  is  — =-. 
baa 


*  The  subject  of  fractional  and  negative  exponents  will  lie  filly  investigated  farther  in 
advance. 


DEFINITIONS  AND  NOTATION.  5 

XV.  The  following  characters  are  used  to  connect  several  quantities  to 
gether.  viz. : 

vinculum,  or  bar     

parentheses  (  ) 

braces,  or  brackets  )  >  or 


Thus,  m-\-?i.x,  or  (m-\-n)x  signifies  that  tho  quantity  denoted  by  m-f-n  is 
to  be  multiplied  by  .r,  and  $  2+jj  I  .  $  £— p-  I  signifies  that  ^+?  is  to  be  multi- 
plied by  ^—-.     The  vinculum  or  bar  is  sometimes  placed  vei'ac-ly  ;  thus, 

-{-ax 

+  c 
signifies  that  the  sum  of  a,  b,  and  c  is  multiplied  by  x. 

XVI.  The  signs,  .-.  therefore  or  consequently,  and  •••  because,  are  used  to 
avoid  the  frequent  repetition  of  these  words. 

XVII.  Every  number  written  in  algebraic  language,  that  is,  by  aid  of 
algebraic  symbols,  is  called  an  algebraic  quantity,  or,  an  algebraic  expression. 

Thus,  3a  is  the  algebraic  expression  for  three  times  the  number  a  ;  5a2  is 
the  algebraic  expression  for' five  times  the  square  of  the  number  a  ;  7ar'b3  is 
the  algebraic  expression  for  seven  times  tho  fifth  power  of  a  multiplied  by  the 
cube  of  b. 

3a2—6b5c4  is  the  algebraic  expression  for  the  difference  between  three 
times  the  square  of  a  and  six  times  the  cube  of  b  multiplied  by  the  fourth 
power  of  c. 

2a—3¥c3-x-Ad4etfs  is  the  algebraic  expression  for  twice  a,  diminished 
by  three  times  the  square  of  b  multiplied  by  the  cube  of  c  and  augmented  by 
four  times  the  fourth  power  of  d  multiplied  by  the  product  of  the  fifth  power 
of  e  and  the  sixth  power  of/. 

XVIII.  An  algebraic  quantity,  which  is  not  combined  with  any  other  by 
the  sign  of  addition  or  subtraction,  is  called  a  monomial,  or  monome,  or,  a  quantify 
of  one  term,  or  simply,  term.  Thus,  3a2,  4b",  6c,  are  monomials.  The  de- 
gree of  a  term  is  the  number  of  its  literal  factors,  and  is  found  by  adding  to- 
gether the  exponents  of  all  the  letters  contained  in  the  term.  Thus,  5a3b2c 
is  of  the  sixth  degree. 

An  algebraic  expression,  which  is  composed  of  several  terms,  separated 
from  each  other  by  the  signs  -(-  or  — ,  is  called  generally  a  polynomial,*  or  poly- 
nome.  Thus,  3a2-r-462— 6c-\-d  is  a  polynomial.  A  polynomial  is  said  to 
be  homogeneous  when  all  its  terms  are  of  the  same  degree. 

A  polynomial,  consisting  of  two  terms  only,  is  usually  called  a  binomial ; 
when  consisting  of  three  terms,  a  trinomial.  Thus,  a-\-b,  3b*c — xz,  are 
binomials,  and  a-\-b—c,  3?rc2/i5— 6p3r+9d,  are  trinomials. 

XIX.  Of  the  different  terms  which  compose  a  polynomial,  some  are  pre- 
ceded by  the  sign  -\-,  others  by  the  sign  — .  The  former  are  called  additive, 
or  positive  tortus,  the  latter,  subtractive,  or  negative  terms. 

The  first  term  of  a  polynomial  is  not,  in  general,  preceded  by  any  sign ;  in 
that  case  the  sign  -\-  is  always  understood. 

*  A  polynomial  is  also  called  a  compound  quantity.  Polynomials,  to  save  the  trouble  of 
writing  them  repeatedly,  are  often  represented  by  a  single  large  letter.  Thus,  if  we  have 
two  polynomials,  x*—  i3py-\-ixy*—y*  and  x*—3xy--\-3x"-y—y3,  we  may  represent  the  first 
by  A  and  the  second  by  B,  and  afterward,  in  referring  to  them,  may  call  them  the  poly 
nomials  A  and  B. 


6  ALGEBRA. 

Terms  composed  of- the  same  letters,  affected  with  the  same  exponents,  are 
called  similar  terms. 

Thus,  lab  and  3ab  are  similar  terms,  so  are  6a2c  and  7a2c;  also,  lOatfrd 
and  2ab3c4d ;  for  they  are  composed  of  the  same  letters,  and  these  letters 
in  each  are  affected  with  the  same  exponents.  On  the  other  hand,  8ab3c 
and  3a~bsc  are  not  similar  terms,  for,  although  composed  of  the  same  letters, 
these  letters  are  not  each  affected  with  the  same  exponent  in  each  term. 

XX.  The  numerical  value  of  an  algebraic  expression  is  the  number  which 
results  from  giving  particular  values  to  the  letters  which  compose  the  ex- 
pression, and  performing  the  arithmetical  operations  indicated  by  the  algebraic 
symbols.  This  numerical  value  will,  of  course,  depend  upon  the  particular 
values  assigned  to  the  letters.  Thus,  the  numerical  value  of  2a3  is  54  when 
we  make  a =3,  for  the  cube  of  3  is  27,  and  twice  27  is  54.  The  numerical 
value  of  the  same  expression  will  be  250  if  we  make  a =5;  for  the  cube  of  5 
is  125,  and  twice  125  is  250. 

The  numerical  value  of  a  polynomial  undergoes  no  change,  however  we 
may  transpose  the  order  of  the  terms,  provided  we  preserve  the  proper 
sign  of  each.  Thus,  the  polynomials  4a3 — 3a26+5ac2,  4a3+5ac2 — 3a2b, 
5ac2 — 3a"b-\-4a3,  have  all  the  same  numerical  value.  This  follows  mani- 
festly from  the  nature  of  arithmetical  addition  and  subtraction,  for  it  is  evident 
that  if  the  same  amounts  be  added  or  taken  away,  it  is  immaterial  in  what 
order. 

Examples  of  the  numerical  values  of  algebraic  expressions  : 

Let  a=4,  fe=3,  c=2 ;  then  will 

(1)  a  +  b—c=4<  +  3— 2=7— 2=5 

(2)  a2+a  Z>+Z>2=42+4  X  3  +  32=16+12+9=37 

(3)  ac— a  5  +  6  c=4x  2— 4X3+3X2=8— 12+6=2 
a2+Z>2— c2  4-+32— 22  16+9—4      21 


(4) 


ab—ac+bc~4x3— 4x2  +  3x2      12— 8+6~10 


(5)  V(«  +  ^-V(«-^)c3=V(4  +  3)X2-V(4-3)x23=Vl4-V8 

=  3-7416574  —  2  =  1-7416574 
a+&     a—c     a  —  b_7_     2     1     263 

(6)  ^H7+6  +  c— a  +  6— 2+5— 7—  70 

XXI.  Entire  quantities  are  those  which  are  rational  and  contain  no  de- 
nominator; such  are  47,  2a26,  3a2 — be. 

XXII.  An  algebraic  expression  containing  a  quantity  is  called  a  function  of 
that  quantity.     For  example,  the  expression  3.r2 —  ^/x  is  a  function  of  a: ;  the 

expression  a(s:-\-y)-\-—{x-{-y)  is  a  function  of  x-\-y.     An  entire  function  of 

a  quantity  is  one  in  which  this  quantity  does  not  enter  into  a  denominator. 

A  rational  function  is  one  in  which  the  quantity  does  not  appear  under  a 
radical. 

To  express,  in  a  general  way,  a  function  of  x,  we  write  F(x).  Where 
many  different  functions  of  x  are  to  be  represented,  we  vary  the  form  of  this 
initial :  thus,  F(x),  f(x),  <p{x),  F'(.r),  &c,  which  denote,  in  a  general  way, 
different  algebraic  expressions  containing  x. 

To  express  functions  of  the  same  form  of  different  quantities,  we  use  th«» 
same  initial  before  these  quantities;  thus,  F(.r),  F(y). 


REDUCTION  OF  TERMS. 


To  express  a  function  like  x~-\-2xij-\-if  of  two  quantities,  we  write  F(x,y\,- 
of  three  quantities,  F(.r,  y,  z),  and  so  on. 
What  follows  to  equations  may  bo  called  the  algebraic  calculus. 


REDUCTION  OF  TERMS. 

3.  Reduction  of  similar  terms  is  the  collecting  of  several  similar  terms  into 
one. 

The  rule  may  be  dividod  into  two  cases : 

(1)  When  the  similar  quantities  have  the  same  signs. 

(2)  When  the  similar  quantities  have  different  signs. 

CASE  I. 

When  the  similar  quantities  have  the  same  signs. 
Add  the  coefficients  ;  affix  the  letter  or  letters  of  the  similar  terms,  ana 
prefix  the  common  sign  -4-  or  — .* 
Thus,  a+2a+3a+4a+5a  =  (l+2+3+4  +  5)a=15a, 

(_a)-f-(— 2a)  +  (— 3a)?f(— 4a)  =  — (l  +  2+3+4)a  =  — 10a. 
It  is  convenient  to  write  the  similar  terms  to  be  reduced  under,  instead  of 
after  one  another,  they  being  read  in  the  same  order  in  either  way. 


(1)  (2)  (3)                        (4)  (b)_ 

3a  abc  9axy  —  bbx  V a-\-x 

la  2abc  •           3axy  —  2bx  2^/a-\-x 

2a  7abc  laxy  —     bx  b^a-\-x 


EXAMPLES. 

(3) 

(4) 

9axy 

—  bbx 

3axy 

—  2bx 

laxy 

—  bx 

baxy 

—  bx 

axy 

—  Abx 

5axy 

—lObx 

\ 

a  3abc  baxy  —     bx  -\/a-\-x 

6a  abc  axy  —  Abx  7-<Ja-\-x 

8a  babe  baxy  — lObx  4  y/a-\-x 

27a  19abc 


CASE  II. 

When  the  similar  quantities  have  different  signs. 
Collect  into  one  sum  the  coefficients  affected  with  the  sign  -f-»  and  also 
those  affected  with  the  sign  —  ;  to  the  difference  of  these  sums  affix  the  com- 
mon literal  quantity,  and  prefix  the  sign  -|-  or  — ,  according  as  the  sum  of  the 
-f-  or  —  coefficients  is  the  greater.f 

*  The  truth  of  this  rule  is  evident ;  for  suppose  the  two  terms  3a  and  5a  are  to  be  r©« 
duced  to  one,  then  by  the  definition  of  a  coefficient  we  have 
5a-=a-\-a-\-a-\-a-\-a 
3a=a-\-a-\-a 
Hence  5a-\-3a=za-\-a-\-a-\-a-\-a-^-a-\-a-\-a=8a. 

Similarly,—  5«=(— «)+(— «)+(— *)+(—«}+(—  «) 

-3a=(-a)-H-a)+(-a). 
Hence -5a+(-3a)=(-a)+(-a)+(-a)+(-a)+(-a)+(-«)+(-«)+(-a) 
=8(— a)=— 8a. 
t  The  truth  of  this  will  be  obvious  ;  for  to  redaoe  5a  and  — 3a,  we  have 
5a=-a-\-a-\-a-\-a-\-a 
-3a=(-«)+(-«)+(-a). 


ALGEBRA. 

Thus,  a  — 2a  +  3a  — 4a  +  5a  =  (l-f3+5)a  —  (2+4)a  =  9a— 6a=3a 
And,  3x+4y— 2x-\-3y=(3— 2)x+{<L  +  3)yz=x+7y. 
Reduce  the  terms  of  the  polynomials, 

(6)  c+2d— 2c— 3d+3c+ld— 4c— 5d+c-{-d 

(7)  3a— 26+5a  — 6e+36  —  9c+a— 6-fl21c 

a     m     k      m      a      k     m 

(8)  *— J—  5—  oo  —  7— 13 — 8  —  9 

(9)  3a— |6-f  6a— 3§6-fl0ia— 22£6— fa 

(10)  bxy—A  fypqr+4xy— 10a563-f7  Vpqr— 9:n/+3a5&*. 


ADDITION. 
Addition  is  the  collecting  of  several  polynomials  into  one. 

RULE. 

Write  the  polynomials  one  after  another,  and  reduce  similar  terms.* 

EXAMPLES. 

f)        

20   (a2— fry— 15  -y/x2— 3/s 

-/a2— 62    —  7  Vx*—y 

12  ja°  —  b'2    —      ^x-—y'i 

4    (a2_&2)*_  3    (x2— i/2)* 

2   (ag__ft3)i_  5   (x2— 3/2)1 


(1) 

(2) 

3a2-f-     62 

2x2 —  xy 

2a2 +  362 

4z2— 7x?/ 

6a2+  5&2 

3x-—4xy 

a2+  762 

x2 —  xy 

a2+  662 

8a;2 — Ixy 

13a2-f2263 

(4) 

(5) 

a+  6 

xy —  ab 

-2a+36 

2xy-\-3ab 

3a— 46 

— 5xy-\-7ab 

-5a+66 

—  xy — 3a6 

7a—  & 

8xy — 9a  b 

4a -4-56 

& 


(6) 


■\/x"-\-y'i  —     vi"-\-    ri1 — 2mn 

— 2  Vx^+y  +  3m2— 3  ?i2+5*n7i 

— 5-/a-'2+2/2  —  4/?i2+5  n2 — Imv 

2  (x2+2/2)-4-12»i2— 2|n2+  mn 

8  (x2-j-?/2)-  —  8m2—  jra2— 6mn 


In  example  (4),  let  a=5  and  6=3,  then  a-\-  b=     8 

_2a+36=— 1 

3a— 46=     3 

— 5a+66  =  — 7 

7a—  6=   32 

4a+56=   35f 

Hence  5a+( — 3a)=a+a+a-fa+a+( — a)+( — «)+( — a). 

=<z4"a=2a- 
Similarly,  2a+(—5aJ=a+a+(—o)+(—o)-{-(—o)+(—o)+(— a) 

=+(-«)+(-«)+(-«) 
s=3{ — a)= — 3a. 
•  For  if  certain  quantities  are  to  be  added  and  subtracted,  it  is  immaterial  in  what  por* 
tions,  or  what  order. 

t  Similar  substitutions  maybe  tried  in  some  of  tbe  following-  examples.  Let  the  learner 
substitute  any  other  numbers  for  a  and  b,  and  he  will  find  that  the  sum  of  the  polynomials 
will  be  truly  expressed  by  the  result  4a-\-5b,  the  correctness  of  which  does  not  depend  od 
the  values  of  a  and  b.    This  illustrates  the  general  principle  stated  in  the  note  of  Art  I 


ADDITION 
(7)  (8) 


bax*    —   \Af+;y    -f-      (a  —  ty  2y/xy-\-xz-\-yz    -f   ty  ax-\-by 

~-  7aJx+2   (a+y)*—  3(a  — b)  —  5 -/•"/+•*-  +  ?/:    —3  {ax+by)* 
12aVx—3Vx+y    +  12{a  —  6)  12    (xy+xz+yz)^ + 5   (a.r+%)* 

—  3a -/a'— 4  Vi-|-i/    —     (a— b)  —  3  -v/ar^+zz  +  i/z     —  2y<ix-\-by 

—  ax*      +      (ar-j-y)*—  3(a  — &)  (xy+xr+yz^-f-      (ax-t-fcy)* 


(9)  (10) 


a+6+c+rf+c— /  4(a+&)Vx2— y3    —  2(a  —  fc)  y/x*+y* 

a+b+c+d  —  e+f  —  3(a+b)Jx*—y2    +  (a  —  6)  V^+y* 

a+b  +  c-d+e+f  -     (a+b)    (x*-if)*+3(a-b)    (x2+y2)i 
a+b-c+d+e+f  6(a+b)    (x3-y2)*-  (a-b)    (x2+y3)i 

a  —  b  +  c  +  d+e+f  10(a+Z>)Vz2— t    —b{a  —  b)    (x2+y2)* 

—a+b+c+d+e+f  —  2(a+6)    (x3— iff+4{g— b)  Vx"*+y* 


4.  Dissimilar  quantities  can  only  be  collected  by  writing  them  in  succession 
and  prefixing  to  each  its  respective  sign.     Thus,  9xy,  — bed,  and  3ab  are  dis 
Mmilar  quantities,  and  their  sum  is  9xy-\-3ab — bed.     In  like  manner,  2ab, 
3aZ>2,  4a&3  are  dissimilar  quantities,  and  their  sum  is  2ab-\-3ab2-\-4ab:i ;  which, 
however,  admits  of  another  form  of  expression,  as  will  be  explained  in  the  rule 
of  Division.     When  several  polynomials,  containing  both  similar  and  dissimilar 
quantities,  are  to  be  collected  into  one  polynomial,  the  process  of  addition  will 
be  much  facilitated  by  writing  all  the  similar  terms  under  each  other  in  verti 
cal  columns. 

This,  however,  is  not  absolutely  necessary.     The  similar  terms  may  be  col 
lected  together  as  they  stand. 

EXAMPLES. 

(1)  Add  together  ax  +  2by  +  cz;   -/•?+  Vy+  V~;  3y-  — 2xi-{-2zi;   4c 
—  3ax«—2by;  2ax—4-\/y—2z^. 

ax+2by+cz  + -/x-f- -/y  +  Jz 
—  3ax— 2&y+4cz  — 2x*4-3y*    +3z4 
2ax  —4  Vy— 22* 

5c; —  \/ .r-|-2-v/ 2=  sum  required. 

(2)  Add  together, 

4a26  +  3csd— 9m"n  ;  4m*n  +  ab"  +  bcH  +  7a"b  ;  6m"n—bc3d  +  'imni—8aba ; 
Imv?  -f  QcH— bm?n  —  6a26  ;  IcH  —  10a62— 8m2n  —  10d4  ;  and  12a26 — 6a?*8 
-\-2czd-irmn. 

A.rranging  the  similar  terms  in  vertical  columns,  we  have 
4a26  +   3c3d—  9m?n 
7a26-f   bcU-\-  4m2n+     a?;3 

—  b<?d-\-  §m?n—  8a62-|-  4mn,2 
—  6a-b-\-   6c3d—  bm"n  -\-  7?/m2 

+   7cM—  Qm?n— 10a&2  —10c/4 

12a2Z>  +   2c3o7 —   %ab°"  -\-mn 

17a26-f  18c3d—  12m-n— 23ab2+  llwm3— 10di+?nn=  sum. 

(3)  Add  ll&c-j^ad— Sac+Sca7;  8ac+7bc—2ad-\-4mn;  2cd—3t:b+bat 
f  a»;  and  9an — 2bc — 2ad-\-bcd  together. 


(7)  Add   ^_3_^+6J^_%±l)  and   ?a     ^__12Vp     4^+r)    to 
v/  y        c     '      z  s  y   '     c  z  s 


10  ALGEBRA. 

(4)  Add  together,  without  arranging  the  similar  terms  in  vertical  columns, 

2ab2-\-3acz—  8cx2  +  9b2x—  8hy2—10ky 
5a3  —lab2—  7bx2—     b2x—  Aky2—\bhy 
Sky—  htf+Jlx    +  14&3  —  22ac2— 10.E2 
19ac2  —  8&2x+   9z2  +   6%+  2/fy2+  2ab2 
ba3  —  8cx2—     x"  +  11.T   —  9hy2+Ub3—2ky2—bky  —  9hy—7bx2. 

(5)  Add  together  a3— b3  +  3a2Z>  —  5a62  ;  3a3— 4a26  +  3&3— 3aZ>2 ;  a3  +  o3 
+  3a2Z>;  2a3— 4b3— bab2;  6a2b+10ab2  ;  and  —  6a3— 7a2b  +  4ab2+2b3. 

(6)  Add  V^+t/2—  jx2—y2  —  bxy  ;  —  3(x2  —  y2f  +  8.ry — 2{x2  +  y2)h  , 
2 -y/^+i/2— 3a:2/  —  5  Vx2—y2 ;  7 .1-3/  +  10  jx2—y2  —  I2^x2+y2  ;  and  rrj 
-j-  *J  x2 — y2-\-  -\/x2-{-y2  together. 

(?)  1 
gether. 

ABA  B 

(8)  Add  together  4A— 6 — \-7^  and  7-— 2A+3^. 

(9)  Add  together  3  cos  a— A  sin  6  +  6  tan  c,  2  cos  a+2  sin  6  +  7  tan  c 
aDd  cos  a+3  sin  & — 2  tan  c. 

(10)  Add  together  3.290  —2.45  D  +1.84  W,  4.560  +0.59  ])  +6.41  tf 
and  2.220+3.11  D  —  4.21  W. 

ANSWERS. 

(3)  16oc+5ac+12ci+4mra— 3a&  +  10aw. 

(5)  a3+a2&+a&2+&3. 

(6)  2  -/a-3— 1/2— 10  V*2+2/2+8:n/. 
13a     5m3     6^/0     (f+r) 

(7)  + yjL+nju.. 

v  '     y     '     c  2      '        s 

A         B 

(8)  2A+-+10^. 

(9)  6  cos  a+sin  6+11  tan  c. 
(10)  10.07© +1.25  D +4.04  W. 

5.  When  the  coefficients  are  literal  instead  of  numerical,  that  is,  denoted  by 
Setters  instead  of  numbers,  their  sum  may  be  found  by  the  rides  for  the  addi- 
tion of  similar  and  dissimilar  terms ;  and  the  sum  thus  found  being  enclosed  in 
a  parenthesis,  and  prafixed  to  the  common  literal  quantity,  will  express  the 
Bum  required. 

EXAMPLES. 

(1)  (2) 

ax+by+cz  3az+   (a+&)  (x+y)+2mnz2 

bx+cy+az  _ax+2(a+&)  {x+y)—bmnz2 

cx  +  ay+bz  4m«z2+5(a+Z>)  (x+y)  +  10ax 

2po22+   (p+g)  (x+y)  +  2px 


(a+6  +  c).r 


+  (&  +  c+a)y  \  =-«um.  {12a+2p)x+  ^8{a  +  b)+p+q^{x+y)  ( 
+Jc+fl+o)zj_  +  (Wm  +  2/;ry)z' 


sum. 
1 


SUBTRACTION.  11 

m 

(3)  (4) 

{a     b)  Vz  +   (»»— n)y/y  +    J2  (m+n)  f  —  {  a—  b)x*+axy 

(a+c)      x'J—  (m— n)     2/4+2-/2  (»  —  p)  I/8— (24+  b)x"-—bxy 

(b—c)Jx  +3(m  —  n)  Vy  —  3-/2  (p— 2w)#2— (  c— 3a)x3+cxy 

( c- a)  y,r  —  5(m— n)  Vy  —6  -/2  (?—>»)  t~  (  c4-2a>2— axy 


(5)  Add  ax3+%  +  c  to  ii'2+%4-^- 

(6)  Add  together  x2-f.ry + i/2;  ax2— axy+ay*2;  and  —  i?/24-&x?/4-&x». 

x24-.rw4-')/2       ,  x-—xy-\-y"- 
(?)  Add  K*+2/)  »nd  \{*—y)'     AIs0'  o  and  ~       2 

(8)  What  is  the  sum  of  («4-&)x4-(c— r%  — re-/ 2;  (a  —  &)x4-(3c4-2a*)v 
4-5x-/2;  26x-f3%  — 2x-v/2;  and  —3bx—Jy— 4x^21 

(9)  Add  ax 4-% 4- cz  ;  a'x— t'y-f-c'z  ,•  and  a"x-\-b"y— c"z. 

(10)  Add  together  ax-\-by 4- cz;  a1x4-^1i/ — cxz  ;  and  a2x — &22/+c22 

ANSWERS. 

(3)  (a4-c)  jx— 2(m  —  n)^y— 6^/2. 

(4)  o?/3— (2c4-2<i)x34-(a-rJ-f  c  —  a^n/. 

(5)  («4-4r24-(64-/0i/+r4-^. 

(G)  (i4-a4-fc)x34-(l_a4-i).n/4-(14-a-%*. 

(7)  First  part,  x.     Second  part,  x34-y2. 

(8)  (2a  — i)x4-(4c4-3o7)i/— 2X-/2. 

(9)  (aJ-a'+a")x-\-(b  —  b'+b")y+(c+c,—c")z. 
(10)       a 

-fa, 
+«3 


x4-6 

y+c 

+  &, 

— c, 

-&„ 

+  Co 

SUBTRACTION. 

ROLE. 

6.  Place  the  quantity  to  be  subtracted  under  that  from  which  it  is  to  be 
taken ;  change  the  signs  of  all  the  terms  in  the  lower  line  from  4-  to  — ,  and 
from  —  to  4~)  or  else  conceive  them  to  be  changed,  and  then  proceed  as  di 
rected  in  Addition.* 

*  The  sign  — ,  prefixed  to  a  monomial,  serves  to  intimate  that  tins  monomial  ought  to  en 
ter  subtractively  into  any  combination  of  which  it  forms  a  part.  If,  for  example,  it  be  r© 
quired  to  add  the  subtractive  quantity  ( — d)  to  c,  the  sum  c-\-[ — d)  is  c — d. 

If  the  difference  between  two  quantities,  as  m,  and  s,  be  required,  m  and  s  being  both  add 
itive,  the  expression  of  the  difference  is  m — g.  If  the  difference  be  required  between  m 
an  additive,  and  ( — s),  a  subtractive  quantity,  let  the  difference  =d ;  that  is,  let 

m — ( — s)=d. 

Adding  ( — s)  to  both  these  equals,  there  results 

But  m — ( — s)-\-{ — s)=m,  and  d-{-( — s)=d — s. 

Therefore,  m=d — s. 

Now  vi — ( — s)=d,  and  m=d — s. 

Hence  m — ( — s)  is  greater  than  m  by  the  additive  quantity  s  cr  is  equal  to  »»4~s 

The  above  is  the  demonstration  for  isolated  temis. 

For  polynomials  we  have  the  following: 

It  is  evident,  that  if  all  the  terms  of  the  quantity  to  bo  subtracted  are  affected  with  tha 


ly  ALGEBRA. 

EXAMPLES. 

(1)  (2) 

From4a+36— 2c+8d  From      :2xy+3y'1  — m2+3  y/2 

Take    A+2&  +   c-\-bd  Take   —   5xy+7y'2-19x'i+2  a/2 

Rem.  3a+   6  —  3c+3gZ  Rem.       17xy— 4y24-   2a:2 +    -y/2 

(3)  (4)  (5) 

32a+   36  28aa:3— 16a2x2+25a3:r— 13<z4         2{a  +  b)+3(a—x) 

5a+17i  18ar5+20a2x2— 2ia3x—  la*  (a  +  b)—3{a—x) 


(6)  (7)  

Gaby— Zyx+izx  ^/xi— 2/'3+4(.?:  -\-y  )  — Sa^H-* 

— 2gfy+6z:r+2;j/3:  3(x  +  ;/)  — 2(.r2— ,y2)"+3    («+*)* 


(8)  (9) 

x2+2a:7/+i/a  «2— 2xy+y<i-\-{x*— y*)  +  2(xy— y*) 

x*—2xy-\-y'i  x2-\-2xy—y"-\-{xi-{-y")—2{xy—yi) 


(10) 
2a2+  ax+  xi—12aix-\-20ax'2—  Ax3  +  6a2x2— Wax3 
g2—3ax-\-2x'2—16a'ix-\-12ax0-—l2ax3—<ix3    -f-  2a2:r* 


(11) 


4y<2-.4yx-\-x'2—2a(x+y)-\-  6 -/a2— x2— 8  Vfc2— y2 
4a:2— 4xy4-y2— 4g(a:4-3/)— 10  \7fr2— 1/2+4  Va2— z2 


7.  In  order  to  indicate  the  subtraction  of  a  polynomial,  without  actually  per- 
forming the  operation,  we  have  simply  to  inclose  the  polynomial  to  be  sub 
tracted  within  brackets  or  parentheses,  and  prefix  the  sign  — .     Thus,  2d 

sign  -}->  we  must  take  away,  in  succession,  all  tho  parts  or  terms  of  the  quantity  to  be  sub 
tracted;  and  this  is  indicated  by  affecting  all  its  terms  with  the  sign — .  But  if  some  of 
the  terms  of  the  subtrahend  are  affected  with  the  sign  — ,  as,  for  instance,  if  c — d  is  to  be 
subtracted  from  a-\-b ;  then,  if  c  be  subtracted,  we  shall  have  subtracted  too  much  by  d , 
hence  the  remainder  a-\-b — c  is  too  small  by  d;  and  therefore,  to  make  up  the  defect,  the 
quantity  d  must  be  added,  which  gives  a-\-l — c-\-d ;  by  inspecting  which  we  perceive  thai 
the  signs  of  the  subtrahend  have  been  changed. 

This  reasoning  may  be  generalized  by  supposing  c  to  represent  the  sum  of  the  additivt- 
terms,  and  d  to  represent  the  sum  of  the  subtractive  terms  of  the  lower  line,  or  quantity  tr 
be  subtracted 

Another  mode  of  proving  the  rule  for  the  signs  in  subtraction  is  the  following : 

By  subtraction  we  solve  the  problem,  "  Given  one  of  two  quantities,  and  their  algebraical 
gum,  to  find  the  other." 

Let  A  be  any  algebraical  quantity,  simple  or  compound,  from  which  it  is  proposed  to 
eubtract  another  simple  or  compound  quantity,  B.  The  quantity  A  may  be  conceived  to  be 
the  algebraical  sum  of  B,  and  some  other  quantity  which  it  is  proposed  to  discover.  Call 
it  X.  As  A  was  obtained  by  annexing  to  x  the  polynomial  expressed  by  B,  with  its  prope? 
signs,  the  effect  of  this  process  will  be  destroyed  by  annexing  to  A  the  polynomial  repro 
tented  by  B,  with  its  signs  changed. 


MULTIPLICATION.  J3 

— 3a26-f4a6-  — («3+634-a62)  signifies  that  the  quantity  a3-f-63+a6'  is  to  bo 
subtracted  from  2a3 — 3a-b-\-4ab2.  When  the  operation  is  actually  perform- 
ed, we  have  by  the  rule, 

2a3— 3a26  +  4a62— (a3+63+a62)=2a3— 3a26  +  4a62— a3— 63— ab* 

=  a3— 3a26+3a62— 63. 
When,  therefore,  brackets  are  removed  which  have  the  sign  —  before  them, 
the  signs  of  all  the  terms  within  the  brackets  must  be  changed. 

8.  According  to  this  principle,  we  may  make  polynomials  undergo  several 
transformations,  which  are  of  great  utility  in  various  algebraic  calculations. 

Thus, 

a3— 3a26+3a62— 63=a3— (3a26— 3a62+63) 
— a3—&3— (30*6— 3a&8) 

=a3+3a62— (3a26  +  63) 
=  — (— a3+3a26— 3a62-f63) 
And  x2—2xy+y"=X'—(2xy—y-)=y'2—(2xy—x'2). 

EXAMPLES    OF   QUANTITIES  WITH   LITERAL   COEFFICIENTS. 

(1)  (2) 

From          ax't+byx+cy2  From  (a+i)  V^2+?/2+(a  +  c)(<z+x)s 

Take  dx^—hxy+ky2 Take  (a— b)  ■s/x"+y'i-\-         c  (a+x)* 

Rem.  (a— ^).T24-(6+/i)^3/+(c— %2.         Rem.  2b  ^x'*+  y3+a{a+x)3. 


(3)  From  m-nV — 2mnpqx-\-2)'1<ii  ta^e  P~q2J? — 2pqmnx-{-m,"n-. 

(4)  From  a(x-\-y)  —  bxy-{-c(x — y)  take  A{x-\-y)-\-{a-\-b)xy — 7(x— y). 

(5)  From  (a-f-6)  (x+y)—{c  —  d)  (x— y)+h*  take  (a  — b)  (x+y)  +  {c+d\ 
(r-y)+£2.  

(6)  From  (2a— 56)  ^x+y-\-{a— b)xy— cz2  take  3bxy— (5+c);2  —  (3a— b\ 
(x+yf. 

(7)  From  2x—y-\-{y—2x)  —  {x—2y)  take  y— 2x— {2y— x)  +  (x+  2y) 

(8)  To  what  is  a+6-fc— (a— b)  —  (b— c)  —  (— b)  equal? 

(9)  From  A^+B^+Car+D  take  A^+B^+C^x+D,. 

ANSWERS. 

(3)  (mW— f-q^X'+pY'— mW,  or  (mV-^jy-^'-pY).  or  {in*n* 

(4)  (a-4)(x+y)-(a+2b)xij+(c+7)  (x-y). 

(5)  2b(x+y)—  2c(x— y)+h'i— Tc*. 

(6)  (5a— 66)  Vx+3/+(a— 4b)xy+5z*. 

(7)  y—x. 

(8)  26+ 2c. 

(9)  (A-A1)r5+(B-B1)x2+(C-C1)x+D-Di. 


MULTIPLICATION. 


9.  Multiplication  is  usually  divided  into  three  cases  : 

(1)  When  both  multiplicand  and  multiplier  are  simple  quantities. 

(2)  When  the  multiplicand  is  a  compound,  and  the  multiplier  a  simple 
uantity. 

(3)  When  both  multiplicand  and  multiplier  are  compound  quantities. 


14  ALGEBRA. 

CASE   I. 

10.  Wlien  both  multiplicand  and  multiplier  are  simple  quantities,  or  monomials. 
To  the  product  of  the  coefficients  affix  that  of  the  letters.* 

Thus,  to  multiply  bx  by  4y,  we  have 

5x4=20;    xxy=xy; 

. •  .bx  X  43/ = 20  X  xy  ==  20x?/  =  product. 

11.  Powers  of  the  same  quantity  are  multiplied  by  simply  adding  their  id 
4ices ;  for  since,  by  the  definition  of  a  power, 

ab—aaaaa  :  a'r=aaaaaaa, 
.'.a5  X  a7=aaaaa  X  aaaaaaa=aaaaaaaaaaaa=al2=a5+7. 

Also,     am=aaa ....  to  m  factors  ;  a°=aaa to  n  factors ; 

.'.am  X  aa=aaa ....  to  m  factors  X  ctaa to  n  factors ; 

=aaaaaa to  (m-{-n)  factors ; 

=am+n. 
It  is  proved,  in  the  same  manner,  that  amXanX«hXak=«ra+n+b+k- 

*  I.  The  rule  is  derived  in  the  following  manner :  We  begin  by  assuming  that  when 
several  letters  are  written  one  after  another  without  any  sign,  their  continued  multiplica- 
tion is  understood,  and  that  the  operation  proceeds  from  left  to  right.  Then  abed  will  sig- 
nify a  multiplied  by  b,  that  product  by  c,  and  that  again  by  d.  We  shall  now  prove  that  in 
whatever  order  these  letters  or  simple  factors  are  arranged,  their  continued  product  will 
always  be  the  same  ;t  and,  moreover,  that  they  may  be  grouped  into  partial  products  at 
pleasure,  provided  all  the  letters  be  employed  each  time.  Thus  the  above  product  may  be 
written  bade  (the  multiplication  here,  as  before,  going  on  by  each  factor  successively  from 
left  to  right),  and  the  result  will  be  the  same  as  before  ;  or  it  may  be  written  aXbXcd,  un- 
derstanding the  products  separated  by  the  sign  X  as  being  previously  formed  and  then 
multiplied  together. 

The  demonstration  depends  upon  three  propositions,  which  we  shall  first  establish  ; 

(1)  .     .  aXb=bXa        ^or  *n  *ue  ^joining  table  of  units  let  b  denote  the  number 

b  of  units  in  each  horizontal  row,  and  a  the  number  of  rows, 

' A" v  then  b  multiplied  by  a,  or  repeated  a  times,  will  give  the 

(.   .   .   1   .   .  number  of  units  in  the  table.    But  a,  which  is  the  number  of 

111111  horizontal  rows,  is  also  the  number  of  units  in  each  column , 

111111  and  b  is  the  number  of  columns  ;  then  a  multiplied  by  b,  or 

\1  1  1  1  1  1  repeated  b  times,  will  produce  the  number  of  units  in  the 

table  again ;  whence  b  multiplied  by  a  is  equal  to  a  multiplied  by  b. 

j.^  In  a  similar  manner,  from  the  adjoining  table,  it  may  be 

(a  a  a  a  a  proved  that 

!  a  a  a  a  a  a.b .  c=a  ,c.b  (2) 

\aaaaa  Also  that  a .  £ .  c=a .  (£c)  (3) 

\a  a  a  a  a  *  ' 

II.  By  (1)  abcd=bacd=  by  (2)  bcad=  by  (2)  beda.  Thus,  we  perceive  that  the  factor 
a  has  been  made  to  occupy  successively  every  place  from  the  first  to  the  last.  The  same 
might  now  be  done  with  the  factor  b,  and  so  with  all  the  others.  Therefore  a  product  is 
the  same,  whatever  be  the  order  of  its  factors. 

III.  Again.  Take  aXbXcXdXe.  It  may  be  written  by  (3)  aXbcXdXe  or  by  (3) 
aXbcdXe,  or,  instead,  by  (3)  abXcdXe.  From  which  it  appears  that  the  factors  of  a 
product  may  be  grouped  into  partial  products  at  pleasure,  and  then  afterward  multiplied 
together  or  conversely. 

IV.  Let  us  now  suppose  that  the  product  3a3£2  is  to  be  multiplied  by  the  prxluct  5a"b*. 
Instead  of  multiplying  by  the  whole  product  5a- J<,  multiply  by  its  factors  separately,  and  we 
have  3a3&25a2j4.  Since  the  order  may  be  changed  at  pleasure,  bring  the  numerical  factors 
together,  and  the  different  powers  of  the  same  letters;  thus,  5 X 3a-a3b*b~.  Grouping  the 
different  powers  of  the  same  letters  into  partial  products,  as  well  as  the  numerical  factors, 
tne  result  is  \5a^b7,  which  has  evidently  been  obtained  by  multiplying  the  coefficients  and 
adding  the  exponents  of  like  letters. 

t  Such  a  relation  as  that  of  a  product  to  its  factors  is  called  a  symmetrical  relation. 


MULTIPLICATION.  J  5 

RULE    OF    SIGNS    IN    MULTIPLICATION.  , 

The  product  of  quantities  with  like  signs  is  affected  with  the  sign  -{-  ;  the 
product  of  quantities  with  unliko  signs  is  affected  with  the  sigu  —  ; 

or 
-f-  multiplied  by  -f-  and  —  multiplied  by  —  give  4-  ; 
-f-  multiplied  by  —  and  —  multiplied  by  -f-  give  —  ; 

or 
like  signs  produce  -f-  and  unlike  signs  — . 
The  continued  product  of  an  even  number  of  negative  factors  is  positive ;  of 
an  uneven  number,  negative.* 

EXAMPLES. 

(1)  4a"b2cd  X  3abc*d*       =       12a3b3c3d3. 

(2)  12  i/ ay  X*bx  =        48bx-/ay. 

(3)  5£tYz*X6ayz3         =       ZZxhf'z'1. 

(4)  13a%»aty  X  —bdbxif  =  —  65a3b4x*y*. 

(5)  —  5.rmv/n  X  —  4xnym       =+  20xm+nym+a. 

(6)  —  20aPii  X  5amincr  =  —  100am+i'6n-hcr. 

CASE    II. 

12.   When  the  multiplicand  is  a  compound,  and  the  multiplier  a  simple 

quantity. 

Multiply  ^ach  term  of  the  multiplicand  by  the  multiplier,  beginning  at  the 
left  hand ;  and  these  partial  products,  being  connected  by  their  respective  signs, 
will  give  the  complete  product,  f 

EXAMPLES. 

(1)  (2) 

Multiply     a--\-ab    +  b2  Multiply        a2—2ab    +6» 

By  4a By  3xy_ 


Product,  4di-\-4a2b-ir4ab3.  Product,  3a-xy —6abxy -f  Ztfxy. 

(3)  Multiply  5mn+3m'2— 2n2  by  I2abn. 

(4)  Multiply  3ax—5by-\-7xy  by  —labxy. 

(5)  Multiply  —  15a2o+3ai2— 1263  by  —  5ab. 

(6)  Multiply  ax3— bx^+cx— d  by  —  x5. 

(7)  Multiply  •/<*+&+  y/xV—if—Zxy  by  —2t/x. 

(8)  Multiply  amxn-\~bmyn — c*ym—daxm  by  xmyn. 

*  Let  m,  mf  be  two  monomial  quantities  whose  product  is  required.  If  m,  mf  are  both  addi- 
tive quantities,  the  product  mm'  is  an  additive  quantity.  This  is  the  case  of  arithmetic. 
If  the  multiplicand  m  is  an  additive  quantity,  and  the  multiplier  m'  a  subtractive  quantity, 
the  expression  »«X( — '">■')  indicates  that  the  multiplicand  m  is  to  be  subtracted  as  many 
times  as  there  are  units  in  m',  or  that  m'  repetitions  of  the  quantity  m  are  to  be  subtracted, 
which  is  expressed  by  — mm'. 

If  m  is  subtractive  and  m'  additive,  — m,  taken  once  is  — m ;  taken  twice  is  — 2m  ;  tak- 
en m'  times  is  — m'm. 

If  m  and  m'  are  both  subtractive,  the  quantity  — m  is  to  be  subtracted  mf  times.  Now 
— m  subtracted  fence  is  -\-m,  twice  is  -f-2m;  and  m'  times  is  -\-m'm. 

t  1st.  Suppose  the  signs  to  be  all  plus.  The  whole  multiplicand  being  to  be  taken  as 
many  times  as  is  denoted  by  the  multiplier,  each  of  its  parts  or  terms  must  be  taken  so 
many  times.  2d.  For  the  case  where  some  of  the  signs  are  negative,  see  the  demonstra- 
tion in  the  •  ext  note. 


/ 


1G  ALGEBRA. 

t  CASE    III. 


13.    When  both  multiplicand  and  multiplier  are  compound  quantities. 
Multiply  each  term  of  the  multiplicand,  in  succession,  by  each  term  of  the 
multiplier,  and  the  sum  of  these  partial  products  will  give  the  complete  prod- 
uct.* 


EXAMPLES. 


(1) 

u-f  b 
a+  b 
a2-f  ab 

+  ab  +  b* 
ai+2ab  +  b2 

cW* 

(2) 
a  +b 
a  — 5 

a*-\-ab 

— ab- 
a^—b2 

-62 

62 

(3)f 
a  —   b 
a  —   b 
a2—  ab 

—  ab  +  b* 
a2—2ab+b* 

(4) 
a  b  -\-cd 
a  b  — cd 

a2-f  i 
a2— 

(5) 

62 

aW+abcd 
—abed — 

a4+: 

2a3b+ 

a262 
a*b°~—2ab3— b* 

a^fr—M* 

a4+S 

lasb—l 

2a  b3—b* 

(6)  Multiply  4a3— ba?b— 8a&2+263by  2a3— 3ab— 4&2. 
4a3_   r>ai0_  8a62+  2b3 

2a2—  3ab  —  4b2 

8a5— 10a*b— 16a362-f-  4a253 

—  12a4i+15a362-f24a263-    6aZ>4 

—  l6a352-f20a2&3+32a&4— 8&5 

8a5— 22a4fc  — 17a3&2-|-48a2fc3+26a&4— 865=  product 

*  1st.  Suppose  all  the  terms  of  the  multiplier  to  be  affected  with  the  sign  -f-.  -  The  mul- 
tiplicand, being  to  be  taken  as  many  times  additively  as  is  denoted  by  the  multiplier,  must 
be  taken  as  many  times  as  is  denoted  by  each  term  of  the  multiplier  separately,  and  the 
separate  results  added  together.  Sd.  When  there  are  both  additive  and  subtractive  terms 
in  the  multiplier  and  multiplicand.  The  rule  for  the  signs  may  be  thus  demonstrated.  Let 
a — b  be  multiplied  by  c — d.     First  multiplying  a  by  c,  the  product  a  — b 

is  ac ;  but  b  should  have  been  subtracted  from  a  before  the  multi-  c  — d 

plication ;  b  units  have,  therefore,  been  taken  c  times  in  the  a,  which  ac — be 

ought  not  to  have  been  so  taken ;  hence  b,  taken  c  times,  must  be  ad — bd 

subtracted,  and  there  results  ac — be  as  the  product  of  a — b  by  c.  ac — be  — ad-\-bd. 

But  the  multiplier  was  c—d  instead  of  c;  therefore  the  multiplicand  has  been  taken  • 
times  too  often;  d  times  the  multiplicand,  which  will  be  of  the  same  form  as  c  times  the 
multiplicand,  viz.,  ad — bd,  must  be  subtracted,  and  the  rule  for  subtraction  is  to  change  the 
signs  of  the  quantity  to  be  subtracted.  The  result  is,  therefore,  ac — be — ad-\-bd ;  com- 
paring which  with  the  given  quantities  we  perceive  that  like  signs  have  produced  -f-  and 
unlike  — .  To  render  the  demonstration  still  more  general,  a  may  represent  the  assem- 
blage of  the  additive  terms  of  the  multiplicand,  and  b  that  of  the  subtractive ;  c  and  d  the 
same  for  the  multiplier. 

t  The  results  in  examples  (1),  (0),  and  (3)  show,  1.  That  the  square  of  the  sum  of  two 
cumbers  or  quantities  is  equal  to  the  square  of  the  first  of  the  two  quantities  plus  twice 
the  product  of  the  first  and  second,  plus  the  square  of  the  second.  2.  That  the  product  of 
the  sum  and  difference  is  equal  to  the  difference  of  the  squares  ;  and,  3.  That  the  square  of 
the  difference  is  equal  to  the  sum  of  the  squares  minus  twice  the  product 


MULTIPLICATION. 


(7)  Multiply  a'b—ab'  by  h'k—htc'. 
a'b—ab' 
h'k—hk' 
a'bh'k—ab'h'k 

—a'bhk'+ab'hk' 


a'bh'k — ab'h'k—a'bhk' '-{-ab'hk' '=  product. 
(8)  Multiply  xm-\-xm-1y-\-xm--y*-\-zm-3y3-\-  &c,  by  x+y. 

£m_|_;rm-J^_j_;rm-2^2_|_xm-3^3_|_ 

x  +2/ 


(9 

(10 

(11 
(12 
(13 
(14 
(15 
(16 
(17 
(18 
(19 
(20 
(21 
(22 

(9 
(10 
(11 
(12 
(13 
(14 
(15 
(16 
(17 
(18 
(19 
(20 
(21 
(22 


xra+i-|-  xmy-\-  xm~ly2-\-  xm~'2y3-\-. 

_|_  xmy-\-  xm~'i/2-(-  xm-2;y3-f-" 

xm+i  _j_  2x™y -f 2xm~1y2 + 2xm~2y3 + . 


Multiply  x2-f  y2  by  x2—y2. 
Multiply  x2-\-2xy-\-y2  by  x — y. 
Multiply  5a4— 2a3&+4a2o2  by  a3— 4a2&+2o3. 
Multiply  x4-r-2x3+3x2+2x+l  by  x2— 2x-fl. 
Multiply  fx2+3ax— ^a2  by  2x2— ax— \a?. 
Multiply  a"-\-2ab-\-¥  by  a2_2ao  +  Z>2. 
Multiply  x2-\-xy-\-y2  by  x2 — xy-\-y2. 
Multiply  x2-\-y2-\-z2 — xy — xz — yz  by  x-f-^+z- 
Multiply  together  x — a,  x — b,  and  x — c. 
Multiply  together  g-{-h,  g-\-h,  g — h,  and  g — h. 
Multiply  together  2>-{-q,  p-\-2q,  p-{-3q,  andp-\-4q. 
Multiply  together  z— 3,  z— 5,  z— 7,  and  z— 9. 
(am— an-\-a2)  X  {am—a). 
{5a5x3—'ibY)X{5a5x3+Abiy5)  as  ex.  2. 

ANSWERS. 

x4 — y*. 

^s-j-a;2?/ — xy2 — y3. 

5a7— 22a6o+12a5o2— 6a463— 4a364+8a2&6. 
x6— 2x3-f-l. 

5x4+|ax3— ^  a2x2-f  f  a3x+|a4. 
a4— 2a262-f64." 
a^+x2i/2+2/4. 
x3-\-y3-\-z3 — 3xj/z. 

x3 — (a-\-b-{-c)x2-{-(ab-{-ac-\-bc)x — abc. 
g*—2g2h2+hA. 

jp4+10p35+35/?2o2+50pa3+2454. 
24_24z3+206z2— 744Z  +  945. 
a?m — am+n+am+2 — am+1  +  an+1 — a3. 
25a10x6— 166Y0. 


When  the  multiplicand  and  multiplier  are  each  homogeneous,  the  product 
will  be  also ;  and  the  degree  of  each  term  of  the  product  will  be  equal  to  the 
sum  of  the  degrees  of  a  term  in  the  multiplier,  and  a  term  in  the  multiplicand. 

This  serves  conveniently  to  verify  the  accuracy  of  the  operation.  It  is  ap- 
plicable in  the  above  examples  to  all  except  the  12th,  20th,  21st,  and  22d. 


18 


ALGEBRA. 


In  multiplying  one  polynomial  by  another,  there  are  always  two  terms  of  the 
total  product  which  are  not  produced  by  the  reduction  of  similar  terms  in  the 
partial  products.  These  two  terms  are  the  term  affected  with  the  highest 
exponent  of  any  letter,  and  the  term  affected  with  the  lowest  exponent.  If 
the  terms  of  the  multiplicand,  multiplier,  and  product  be  arranged  in  the  order 
of  the  powers  of  some  letter,*  as  is  usual,  and  as  may  be  seen  in  the  above  ex 
amples,  then  the  two  terms  in  question  of  the  product  will  be  the  first  and 
last,  the  one  being  produced  by  the  multiplication  of  the  first  of  the  multipli- 
cand by  the  first  of  the  multiplier,  and  the  other  by  the  multiplication  of  the 
last  of  the  multiplicand  by  the  last  of  the  multiplier.  The  first  of  the  multi 
plicand  by  the  second  of  the  multiplier  usually  produces  a  terra  similar  to  that 
which  is  produced  from  the  multiplication  of  the  second  of  the  multiplicand  by 
the  first  of  the  multiplier.  The  same  is  the  case  with  the  first  and  third  of 
each,  the  first  and  fourth,  the  second  and  fourth,  the  third  and  fourth,  and  so  on. 

"When  a  polynomial,  arranged  according  to  the  powers  of  some  letter,  con- 
tains many  terms  in  which  this  letter  has  the  same  exponent,  these  terms, 
after  suppressing  from  them  the  letter  of  arrangement,  may  be  placed  in  a 
parenthesis,  or  in  a  vertical  column  with  a  vinculum  placed  vertically  on  the 
right,  and  the  letter  of  arrangement,  with  its  proper  exponent,  following  after. 
The  polynomial  in  the  parenthesis,  or  vertical  column,  is  to  be  regarded  as  the 
coefficient  of  the  power  of  the  letter  which  follows,  and  is  to  be  operated  with 
exactly  as  we  do  with  a  numerical  coefficient;  i.  e.,  multiply  the  coefficient 
of  the  letter  of  arrangement  in  the  multiplicand  by  the  coefficient  of  the  same 
letter  in  the  multiplier,  and  afterward  add  the  exponents  of  this  letter. 


26 


Multiplicand    <(  — 1 
Multiplier 


26 
+  1 


EXAMPLE. 

a2—  462 
-f  26 
—  1 

a+  863 
—  46s 

a  —  463 
+   1 

Product  of  the 

multiplicand  by 

26 

+  1 


Product  of  the 
multiplicand  by 

—  462 


+  1 


Total  product 
simplified     < 


(      462 

a3— 16b3 

aa+326* 

—  1 

-f  462 

—  863 

+   26 

—  462 

-  2 

+   26 

—  1 

a— 3265 
+  166* 
+  863 
—  46s 


The  letter  chosen  for  this  purpose  is  called  the  letter  of  arrangement 


MULTIPLICATION. 


19 


WO 


-a 

6 


-o 
-r 


+ 

+ 

cs 

w 

iC 

-O 

co 
1 

CO 

1 

1 

1 

rO 

-o 

CO 

CO 

1-1 
1 

1 

1 

U3 

m 

HO 

* 

CN 

CN 

CO 

CO 

1 

1 

1 

1 

<3 

i-i 

■-I 

1 

1 

rO 

* 

CN 

cn 

+ 

+ 

1 

« 

eo 

r-O 

rO 

CO 
I 

co 
1 

1 

*»* 

rO 

HO 

e» 

CD 

CN 

HO 

i— i 

CO 

-^** 

^— ^ 

1 

o 

ci 

+ 

-Q 

r-O 

00 

-tf 

c* 

+ 

1       + 

+ 

« 

~o     <3 

CD     *— 

~© 

I-l 

i-H     '-1 

cn 

+ 

C4 

1 

rO 

^^ 

-2.* 

-O 

CN 
1 

l-H 
1 

r-H     CN 

+      1 

1 

t-O 

-O    ""° 

n 

rO 

-* 

Tt« 

00    ** 

CD 

I 

1 

I        I 

1—1 

1 

1 

1 

1        1 

1 

n 

e 

e 

e  -o 

« 

^~- 

' — > 

<— «  CO 

'"■» 

1  ■ 

1     1 

1 

W 

e* 

^ 

rO 

-o 

-O  . 

OJ 

CN 

3       1 

-S 

« 


co 


a 


u 


CN 


e 

a 


^2 


i-O 

1 

1 

n 

rO 

-o 

-O 

*# 

i— ( 

co 

CO 

l 

+ 

1 

+ 

» 

*r 

r~."l 

i-O 

-O 

CO 

CN 

■3 
H 

-o 
-f 


~3 
EO 


I-l 

rO     rO 

rH 

CN    CN 

+ 

+     1 

rA       ta 

i-O 

r-O     i-O 

CN    i-l 

rC     Tji 

1     + 

1      + 

n 

rt 

WO      r-O 

rO 

Tt<      CN 

CO 

-r 


00 


r-l 
1 

I-t 

i-l 

CN 

1 

rO 

CN 

1 

-o 

CN 

+ 

CN 

1 

+ 

-o 


.§1 


CO 


KO 


r-O 

n 

HO 
CO 


iJ5    .-O 
CO     M" 


~o 
CO 


-O 
CN 

m 


^    1 


.& 


+ 


fO     wO 
<N     <* 


+ 

n 
-O 
CO 


EO 


CN 

CO 


1 

i-O 

1 

CN 

CN 

+ 

+ 

e» 

i-O 

i-O 

1-1 

•* 

'i' 

+ 

+ 

1 

m 

i-O 

rO 

~o 

1 

I-H 
1 

CO 

1 

CO 
1 

1 

-*• 

1 

HO 

iJD 

-o 

-o 

-tfl 

■<3< 

CO 

CO 

CN 
1 

iO 

~> 

1 

1 

1 

1 

1 

CO 

1 

n 

+ 


MULTIPLICATION  BY  DETACHED  COEFFICIENTS. 

14.  In  many  cases  the  powers  of  the  quantity  or  quantities  in  the  multipli- 
cation of  polynomials  may  be  omitted,  and  the  operation  performed  by  the  co- 
efficients alone  ;  for  the  same  powers  occupy  the  same  vertical  columns,  when 
the  polynomials  are  arranged  according  to  the  successive  powers  of  the  letters ; 
and  these  successive  powers,  generally  increasing  or  decreasing  by  a  common 
difference,  are  readily  supplied  in  the  final  product. 


EXAMPLES. 

(1)  Multiply  xt+xty+xyt+y3  by  x— y. 
Coefficients  of  multiplicand  1-f-  1-f-l  +  l 
multiplier     1 — 1 

1  +  1  +  1  +  1 
—1—1  —  1—1 

1  +  0+0+0—1 


so 


ALGEBIL* 


Since  x3X£=£4>  the  highest  power  of  a:  is  4,  and  decreases  successively  by 
unity,  while  that  of  y  increases  by  unity ;  hence  the  product  is 
^-fO-^y+O.iy+O.i!/3 — 2/4=r» — yi=  product. 

(2)  Multiply  3a2+4ax— 5x2  by  2a2— 6ax+4x2. 

3+  4—  5 
2—  6+   4 
6+   8  —  10 
—18— 24  +  30 

+  124-16—20 
6—10—22+46—20 
♦.  Product  =6a4— 10a3x— 22a2x2+46ax3— 20X4. 

(3)  Multiply  2a3— 3a62+5Z>3  by  2a2— 56*. 

Here  the  coefficients  of  a2  in  the  multiplicand,  and  a  in  tho  multiplier,  are 
each  zero ;  hence 

2+0—  3+  5 

2+0—  5 

4  +  0—  6  +  10 

_10_   Q  +  15—25 
4  +  0  —  16+10  +  15—25 
Hence  4a5— 16a3fc2+10a263+15afc4— 2565=  product. 
The  coefficient  of  a4  being  zero  in  the  product,  causes  that  term  to  dis- 
appear. 

(4)  Multiply  x3— 3x2+3x— 1  by  x2— 2x+l. 

(5)  Multiply  3/2— 3/a+^a2  by  yt+ya— |a2. 

(6)  Multiply  ax— 6x2+cx3  by  1—  x+xs— xs+x4. 

(7)  (x3— ax2+6x— c)x(a:2— c?x+e). 

ANSWERS. 

(4)  r5— Sx^+lOx3— 10x2+5x— 1. 

(5)  ^-ay+ia^-^a4. 


(6)  ax — a 
—6 


x2+a 


xJ — a 
—6 
— c 


x'+a 
6 


x5— & 
—  c 


x5+cx7 


Or,  ax— (a+5)x2+(a+fe+c)x3— (a+6+c)x4+(a  +  &+c)x5— (&  +  c)x« 

+  CX7. 

(7)  x5— (a+d)x4+(Z>+aa7+e)x3— (c+6d+ae)x2+(a2+e&)x— ce 


DIVISION. 

15  The  object  of  algebraic  division  is  to  discover  one  of  the  factors  of  a 
given  product,  the  other  factor  being  given ;  and  as  multiplication  is  divided 
into  three  cases,  so,  in  like  manner,  is  division. 

(1)  When  both  dividend  and  divisor  are  monomials. 

(2)  When  the  dividend  is  a  polynomial,  and  the  divisor  a  monomial. 

(3)  When  both  dividend  and  divisor  are  polynomials. 

CASE  I. 

16    When  both  dividend  and  divisor  are  monomials. 
Write  the  divisor  under  the  dividend,  in  the  form  of  a  fraction ;  cancel  like 


DIVISION.  21 

quantities  in  both  divisor  and  dividend,  and  suppress  the  greatest  factor  com- 
mon to  the  two  coefficients. 

17.  Powers  of  the  same  quantity  are  divided  by  subtracting  the  exponent 
of  the  divisor  from  that  of  the  dividend,  and  writing  the  remainder  as  the  ex- 
ponent of  the  quotient.* 

Thus,  a1  ^=aaaaaaa ;  a4=aaaa 

a7      aaaaaaa 

'  '  a*         aaaa 

Generally,     am=aaaa to  m  factors  ;  a"=aaa to  n  factors ; 

&p  =bbbb  to  p  factors  ;  bi  =bbb to  q  factors ; 

aH?      aaa to  m  factors  X  bbb to  p  factors ; 

*  '  an  61      aaa to  n  factors  X  bbb to  q  factors  ; 

z=aaa... to  (m— n)  factors xbbb to  {p—q)  factors, 

_=am-njp-q- 

When  a  quantity  has  the  same  exponent  in  the  dividend  and  divisor,  we  have 

am  .      am 

_am-m_ao.    Dut  —  =  1. 

am  a 

.-.  a°  =  l. 

Hence  every  quantity  whose  exponent  is  0  is  equal  to  3 . 

a?       aaa        1    '  1  t 

a5     aaaaa     aa     a2' 

But  we  may  subtract  5,  the  greater  exponent,  from  3,  the  less,  and  affect 

the  difference  with  the  sign  — ;  hence 

a3  a3      I 

— =a3-5=a-2 ;  but— =— ; 
a5  a5     a? 

1 
a3 

*  The  rale  for  division  follows  from  its  object,  which  is,  having  one  of  the  factors  of  a 
product  given  to  find  the  other.  As  in  multiplication  we  join  together  the  factors  of  a  prod- 
uct without  any  sign,  and  without  regard  to  order,  in  division  we  suppress  from  the  prod- 
act,  i.  e.,  the  dividend,  one  of  the  factors,  i.  e.,  the  divisor,  to  obtain  the  other,  which  is  the 
quotient.  Note. — The  quotient  must  contain  those  factors  of  the  dividend  which  are  not  in 
the  divisor.  Note,  also,  that  dividing  0110  of  the  factors  of  a  product  divides  the  whole 
product.  Thus,  dividing  a&bc  by  a3,  we  divide  the  single  factor  aP,  and  get  a-bc ;  so  to  di- 
vide 16X12  by  8,  we  divide  16  alone,  and  get  2X12  for  the  quotient. 

When  there  are  factors  in  the  divisor  which  are  not  in  the  dividend,  the  quotient  may 
be  expressed  in  the  form  of  a  fraction,  as  has  been  previously  shown  (2,  V.).  Suppressing 
the  common  factors  in  this  case  amounts  to  dividing  both  numerator  and  denominator  by  the 
same  quantity.  That  such  a  division  does  not  alter  the  value  of  the  fraction,  will  be  obvious 
from  the  following  considerations  : 

1.  If  the  numerator  of  a  fraction  be  increased  any  number  of  times,  the  fraction  itself  will 
be  increased  as  many  times  ;  and  if  the  denominator  be  diminished  any  number  of  times, 
the  fraction  must  still  be  increased  as  many  times. 

2.  If  the  denominator  of  a  fraction  be  increased  any  number  of  times,  or  the  numerator 
diminished  the  same  number  of  times,  the  fraction  itself  will,  in  either  case,  be  diminished 
the  same  number  of  times. 

3.  If  the  numerator  of  a  fraction  be  increased  any  number  of  times,  the  fraction  is  in- 
creased the  same  number  of  times  ;  and  if  the  denominator  be  increased  as  many  times,  tho 
fraction  is  again  diminished  the  same  number  of  times,  and  must  therefore  have  its  original 
value.  Hence  both  terms  of  a  fraction  may  be  multiplied  by  the  same  number,  and,  by 
similar  considerations,  it  will  appear,  may  be  divided  by  the  same  number  without  changing 
the  value  of  the  fraction. 

Corollary. — Rule.  To  multiply  a  fraction  by  a  whole  number,  multiply  the  numerator  of 
the  fraction,  or  divide  its  denominator  by  the  whole  number.  To  divide  a  fraction,  divide 
its  numerator,  or  multiply  its  denominator. 


22  ALGEBRA 

Similarly,  -2_=(a+ar)-»;  ]——.  =  {x+y) 


-3 
> 


A-nd  -xzzz^+v2)-3^2— 2/2)-^,  and  so  on. 

So,  also,      —=-5^= — -; 
a3     a3-5     a-2 

a5 
But  -,=a2 ; 

a3 

1 

•••  -=S=aa- 
ar? 

From  this  it  appears  that  a  factor  may  be  transferred  from  the  denominator 

to  the  numerator,  and  vice  versa,  by  changing  the  sign  of  its  exponent. 

EXAMPLES. 

(1)  Write  a?b3c  with  the  factors  all  in  the  denominator. 

cPb(? 

(2)  Write  -rnz  with  the  factors  all  in  one  line,  and  also  all  in  the  denomi- 

nator. 

For  more  of  the  theory  of  negative  exponents,  see  a  subsequent  article. 

18.  In  multiplication,  the  product  of  two  terms,  having  the  same  sign,  is 
affected  with  the  sign  -}-  ;  and  the  product  of  two  terms,  having  different 
signs,  is  affected  with  the  sign  — ;  hence  we  may  conclude, 

(1)  That  if  the  term  of  the  dividend  have  the  sign  -}-»  and  that  of  the  di- 
visor the  sign  -f->  the  resulting  term  of  the  quotient  must  have  the  sign  -|-  ; 
because  +  X  +  gives  +. 

(2)  That  if  the  term  of  the  dividend  have  the  sign  -f-,  and  that  of  the  divisor 
the  sign  — ,  the  resulting  term  of  the  quotient  must  have  the  sign  — ;  because 

-X—  gives  +. 

(3)  That  if  the  term  of  the  dividend  have  the  sign  — ,  and  that  of  the  di- 
visor the  sign  -\-,  the  resulting  term  of  the  quotient  must  have  the  sign  —  • 
because  -f-  X  —  gives  — 

(4)  That  if  the  term  of  the  dividend  have  the  sign  — ,  and  that  of  the  di 
visor  the  sign  — ,  the  resulting  term  of  the  quotient  must  have  the  sign  +• 

RULE   OF  SIGNS   IN   DIVISION. 

+  divided  by  +>  and  —  divided  by  — ,  give  +» 

—  divided  by  -f,  and  -|-  divided  by  — ,  give  —  ; 

or, 

ike  signs  give  -}-,  and  unlike  — ,  the  same  as  in  multiplication. 

+  ab  ,      — ab  ,      — ab  .     4-ab 

±—  =  +  b;  =  +  b;  - =  —  b;  -2—  =— 4. 

-fa         *         —a         '         -f  a  — a 

EXAMPLES. 

(1)  Divide  48a3o3c2a7  by  12a62c. 

48a363c2J     AQaaabbbccd  ,    , 

r- — = — r^—Ti =4aaoca=4a2ocG 

12ab'2c  \2abbc 

150ar'b8cd3 

,      —  16a262c2  ,  ,   , 

(3)  —, =4a2-1i2-1c2-1=4a6c. 


DIVISION.  2J 

I5aimx3nv4n 

( 4  1  •— =5a  2">-mx3n_2n7 /4"-£n = 5amxn v-0. 

v   '    Sa-^Y" 

— 48am6n  7 

(5)  — r— : =—  8ara-Pin-i. 

v  '  —  7a"bcd3xAy:>z6       T  ^ 

(7)  amincr-i-an&nc=ara"r,cr-1. 

(8)  a3m6n+1cr^4-am6nc=a2,n6cr-3. 

(9)  5aP-^3ai,+r5c-1  =  *a-ri_Ic. 

(10)  am-n-J-ap_q=am~n'"p+q- 

(11)  ah-±—  ab=  —  1. 

(12)  —  a6c-r-a6c=—  1. 

(13)     _//n_^_/,m_1. 

a* 

(14)  96a3&W-^84aZ>4c7cZ5=|^. 

(15)  r-53/-nz-i-3-i-a:-73/-mz-P =x22/m-nzp_<1-3. 

CASE  II. 

19.   When  the  dividend  is  a  •polynomial,  and  the  divisor  a  monomial 
Divide  each  of  the  terms  of  the  dividend  separately  by  the  divisor.* 

EXAMPLES. 

(1)  Divide  6a2xY— l2a3x32/6-|-15a4x5?/3  by  3a2xy. 

Qa"x*y6—12aVy6-\-15a4x5y3         „  ..„«,, 
■'        3a3jy ~ = 2*Y  -±axy4+ bd?x?y. 

t     d* 

(2)  Divide  lba?bc  —  20acif-{-5cd2  by  —  babe.  Ans.  —  3a  +  4^— ^. 

(3)  Divide  xn+1— x^+x1^3— xn+1  by  xn.  Ans.  x— x2+x3— x1. 

(4)  Divide  5(a+b)3  —  10{a+by+15(a+b)  by  —  5(a  +  6). 

Ans.  — (a+Z>)2-f  2(a+6)— 3. 

(5)  Divide  12a4i/6— 16a-y4-20a62/4— 28aY  by  ~ 4aY- 

Ans.  —  37/34-4a3/2— 5a2y+7a3. 

CASE    III. 

20.    When  both  dividend  and  divisor  are  polynomials. 

1.  Arrange  the  dividend  and  divisor  according  to  the  powers  of  the  same 
letter  in  both. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the  divisor,  and 
ihe  result  will  be  the  first  term  in  the  quotient,  by  which  multiply  all  the  terms 
in  the  divisor,  and  subtract  the  product  from  the  dividend. 

3.  Then  to  the  remainder  annex  as  many  of  the  remaining  terms  of  the 
dividend  as  are  necessary,  and  find  the  next  term  in  the  quotient  as  before. 

(1)  Divide  a4— 4a3x+6a2x2— 4ax3+x4  by  a-— 2ax+x2. 

a2— 2ax+x2)  a4— 4a3x+6a2x2— 4ax3+x4  (a2— 2ax-\-xi 
a4_2a3x+  a2r2 

— 2a3x-4-5a2x2— 4ax? 
— 2a3x-f4a2x-— 2ax3 

a-x- — 2ax3-\-x4 
a2x2— 2ax3-|-x4 

*  This  rule  follows  from  that  for  multiplication,  which  requires  each  term  of  the  multipli- 
cand to  be  repeated  as  many  times  as  is  expressed  by  the  multiplier. 


24  ALGEBRA. 

Arranging  the  terms  according  to  the  descending  powers  of  x,  we  have 
i2— 2ax+a2)  x4— 4ax3+6a2x2— 4a3r+a4  (x2— 2ax+a2 
x4-— 2ax3-|-  a2x2 

— 2ax3-r-5a2x2 — 4a3x 
— 2ax3+4a2x2— 2a3x 


a2x2— 2a3x+a4 
a2x2— 2a3x-|-a4.* 

(2)  Divide  x4-\-x2yi-\-y*  by  x2+x?/4-2/2. 

x2+2:3/+2/2)  x44-x2?/2-|-2/4  (x2 — xy~\-y'i 

tf-^&y  -}-x2?/2 

— x*y  +y* 

— y?y  — x23/2 — xys 


x*f+xtf+y* 

x2y2-\-xyn-\-y*. 


*  It  has  been  shown  (13)  that  when  the  dividend  (which  is  the  product  of  the  divisor  and 
quotient)  is  arranged  as  directed  in  the  rule,  its  first  term  is  produced  without  reduction  by 
the  multiplication  of  the  first  term  of  the  divisor  by  the  first  of  the  quotient.  Hence  the 
rule  above  for  finding  the  latter.  This  first  term  of  the  quotient  being  found,  and  tho  di- 
visor being  taken  away  from  the  dividend  as  many  times  as  is  expressed  by  this  term,  the 
remainder  must  contain  the  divisor  as  many  times  as  is  expressed  by  the  second  and  re- 
maining terms  of  the  quotient.  Hence  the  remainder  may  be  regarded  as  a  new  dividend, 
and  the  object  being  to  find  how  many  times  it  contains  the  divisor,  it  must  be  arranged  in 
the  same  manner  as  was  the  given  dividend,  and  the  first  step  will  be  the  same  as  before. 
Similar  reasoning  will  apply  to  the  rest  of  the  process. 

Note. — The  arrangement  of  the  terms  is  for  convenience.  The  term  having  the  highest 
or  lowest  exponent  of  some  letter  might  be  selected  from  the  dividend  and  remainders  with- 
out  any  arrangement.  The  operation  must  always,  however,  begin  with  this  term,  as  a 
reference  to  the  last  example  will  show ;  for  if  we  attempt  to  commence  with  the  term 
Scfix2,  the  third  of  the  dividend,  for  instance,  we  perceive  that  this  is  produced  by  reduction 
from  the  term  a^x2  in  the  second  line,  the  term  Acfix2  in  the  fourth  line,  and  the  term  a2x'1 
in  the  sixth.  The  first  of  these  is  produced  by  the  multiplication  of  the  first  of  the  quotient 
by  the  last  of  the  divisor,  the  second  by  the  multiplication  of  the  second  of  the  quotient  by 
the  second  of  the  divisor,  and  the  third  by  the  last  of  the  quotient  and  first  of  the  divisor. 
It  is  not  till  the  first  and  second  terms  of  the  quotient  have  been  found  by  the  rule  above 
given,  that  any  portion  of  the  term  6a°x~  presents  itself  to  be  divided,  or  that  we  can  know 
what  part  of  it  is  to  be  used  as  a  dividend. 

In  the  same  manner,  it  may  be  shown  that  it  would  be  impossible  to  begin  with  the  second 
term  of  the  dividend  4ax%  until  the  first  term  of  the  quotient  has  been  found,  which,  multi- 
plied by  the  second  of  the  divisor,  produces  — 2ax3,  a  part  of  — Aax3,  and  the  subtraction 
leaves  the  other  part  — 2axz,  which  now  we  know  is  the  product  of  the  first  of  the  divisor 
by  the  second  of  the  quotient,  which  latter  we  may  then  find. 

The  first  of  the  divisor  multiplied  by  the  second  of  the  quotient,  and  the  second  of  the 
divisor  by  the  first  of  the  quotient,  usually  produce  the  same  power  of  the  letter  of  arrange- 
ment, and  reduce  together;  the  first  and  third  of  each,  together  with  the  two  second  terms 
of  each,  usually  produce  the  same  power,  and  so  on.  It  is  only  the  first  of  the  divisor  and 
first  of  the  quotient,  or  last  of  the  divisor  and  last  of  the  quotient,  which  always  produce  a 
term  that  does  not  reduce  with  any  other  term. 

N.B. — The  arrangement  may  begin  with  the  lowest  as  well  as  the  highest  power  of  any 
letter,  and  go  on  increasing  instead  of  decreasing.  When  either  of  these  arrangements  is 
observed,  if  the  first  term  of  the  divisor  in  any  part  of  the  operation  is  not  contained  exactly 
in  the  first  term  of  the  remainder,  the  division  is  impossible.  By  varying  the  arrangement, 
therefore,  or  simply  considering  which  terms  would  come  first,  using  different  letters  of  ar- 
rangement, we  may  often  determine  beforehand  by  inspection  whether  the  division  is  pos- 
sible or  not. 


DIVISION.  25 

Another  form  of  the  work  which  has  the  convenience  of  placing  the  quotient 
which  is  the  multiplier,  under  the  divisor,  which  is  the  multiplicand,  is  the 
following. 


Dividend;  x4-\-x2y'2-}-    y 
x4-\-x3y  -\-x~y- 


i'--\-ry-\-y'2,  divisor. 
x- — xy+y*,  quotient. 


•    x3y  +yA 

— x*y  — x-y2 — xy3 

xiyij^xif-^y*  • 

x'2y'2-\-xys-\-y4 

(3)  Divice  a5— a3Z>2+2a2&3— ab*+b5  by  a2— a&  +  &2. 

a*— ab+b*)  a5— a37>2+2a263— a&4+&5  (a3+a26  — aV+a?_ah  ,  6, 

a46  —  2a362-f2a263 
a*b  —  a3Z>2+  a263 


—  a362+  a263— ab* 

—  a3&24-   a?b3—ab* 


-f  65 


Arranging  the  tei-ms  accoiding  to  powers  of  6,  we  get 

— a46-fa5 
b*—ab  +  a*)  ¥— ab*+2a°-b3— a362+a5  (^3+^+fc3_a6  ■  a» 

as63— a86B+a6 

a263— a362+a4& 


_a4fe-fa5. 

The  results  we  have  obtained  in  these  two  arrangements  are  apparently 
different ;  but  their  equivalence  will  be  established  as  follows  : 
(1)  (a2— ab+b2)  (a3+a26— a&2)=a5— a3i2+2a2&3— ab* 

Add  remainder  =  -j-^5 

Proof a^a3Z>2+2a263-a&4+&5 


(2)  (52— ab-\-a")  (b3+a2b)  =¥—a  6<+2a253  — a362+a4& 

Add  remainder  =         — a4fo-}-a5 

Proof 65— a  b4+2a°b3  — a36s-f  a5. 

The  moment  we  arrive  at  a  term  of  the  quotient  in  which  the  exponent  of 
the  letter  of  arrangement  is  less  than  the  difference  of  the  exponents  of  this 
letter  in  the  last  terms  of  the  divisor  and  dividend,  we  may  be  sure  that  the 
division  will  not  terminate.  If  the  divisor  and  dividend  be  arranged  in  the  re- 
verse order,  that  is.  beginning  with  the  lowest  power  of  a  letter,  then  the 
division  will  not  terminate  when  the  exponent  of  this  letter  in  the  term  of 
the  quotient  is  greater  than  the  difference  of  its  exponents  in  the  last  terms  of 
the  divisor  and  dividend. 

Thus  in  tbe  following  example. 


x9 + x~  —  ax5 + ax1 
3?+ 3*+ as5 


3* +3?+ a 


.c5- 


— x8-|-  a-'7 — 2ax54-  ax* 
— x3 —  x1 —  ax4 

2x  —  2ax5-\-2axi. 


26  ALGEBRA. 

The  last  term  of  the  quotient  must  be  x4,  in  order  that,  multiplied  by  a,  tn« 
last  of  the  divisor,  it  may  produce  the  last  of  the  dividend.  If,  therefoi-e,  the 
division  is  not  completed  when  this  term  containing  x4  is  obtained,  it  will  not  be 

EXAMPLES   FOE  PRACTICE. 

(1)  Divide  a2— 2ab-\-b2  by  a  —  b. 

(2)  Divide  a24-4ax4-4x2  by  a+2x. 

(3)  Divide  12x*— 192  by  3.r— 6. 

(4)  Divide  63— 6y6  by  2x'2—2y". 

(5)  Divide  a^Sa^-^Sa^—b6  by  a3— 3a26  +  3a&2— b3. 

(6)  Divide  x34-5.r22/4-5x,y24-2/3  by  x^-^-ixy-^-y2. 

(7)  Divide  x5 — yb  by  x — y. 

(8)  Divide  a4— 64  by  a3+a26+ai2+63. 

(9)  Divide  x3— 9x24-27x— 27  by  x— 3. 

(10)  Divide  x*-|-2/4  by  x+?/. 

(11)  Divide  48x3— 76ax3— 64a2x4-105a3  by  2x— 3a. 

(12)  Divide  ix34-x24-fx4-£  by  ±x+l. 

(13)  Divide  52m.5  —  93m4p—  70m3jp2  +  48m2p3  —  27mp4  by  13m3  —  InPf 
-J-3mp2. 

(14)  Divide  33a363— 77a264+121a265  by  3a2Z>— 7aZ>24-lla&3. 

(15)  Divide  (6p4— 12pa3— 6^3o+12?4)  by  (i? — 9). 

(16)  Divide  (100a5— 440a4&4-235a3&2— 30a=&3)  by  (5a3— 2a*£) 
•  (17)  Divide  (g4— 4g-3/i+6g-2/i2— 4gh*+h*)  by  (h*—2hg+g*). 

(18)  Divide  (37a2m2— 26a3m4-3a4— 14am3)  by  (3a2— 5am+2m-). 

(19)  Divide  (a6— Z>6)  by  (a— 6)  and  (a6+66)  by  (a+6). 

(20)  Divide  (a7— &7)  by  (a— 6)  and  (a7+67)  by  (a+fc). 

(21)  (£_622+27z4)  -r  (i+2z+3z2)  =  l— 6z+928. 

ANSWERS. 

(!)  a  —  b. 

(2)  a+2x. 

(3)  4r3+8x2+ 16x+32. 

(4)  3x*+3x*y*+3y*. 

(5)  a3+3a2Z)+3a&2+Z>3. 

(6)  x+y. 

(7)  .rJ  +  X3?/-|-X27/2  +  X7/3  +  2/4. 

(8)  a—b. 

(9)  x2— 6x+9. 

(10)  x*-x*y+xy*-y3+-^. 

(11)  24x2— 2ax— 35a2. 

(12)  tf+f. 

(13)  4m2— 5mp— 9p3. 

(14)  llai2. 

(15)  6p»— 12a3. 

(16)  20a2— 80a&4-15&2. 

(17)  g*  —  2ghJrh'!: 

(18)  a?— 7am. 

C  a5  +  a42>  4-  a3i2  +  a2i3  4-  a&4  4-  &5,  and 
M<N    <  2fc" 

^    )  a5  —  a46  4-  a3i2  —  a*V>  +  ab*  —  fc*+— rz- 


DIVISION. 

r  as+a^-t-a^+a^+a^+a^+i6,  and 
I20)    j  ae_ G6Z>+a<62  — a3i3+a2&4  — ab>+¥— ^-p. 

EXAMPLES  WITH   LITERAL  EXPONENTS. 

(1)  Divide  2a3n— 6a2n&n+6a.n&2n  — 2&3a  by  aa— 6". 

a°— 6")  2a3n— 6a2n6n  +  6an62B— 263n  (2a?a—4aabn+2ba* 
2a3n— 2a2n6u 

_4a'2"6"+tJa"^n 
_4a2n6n-(-4a"^;:" 


27 


2a"i2"—  263u 
2ani2n—  2&3n. 

(2)  Divide  xm+1-\-xmy+xym+ym+l  by  xm+2/m- 

(3)  Divide  an— x"  by  a— x. 

(4)  Divide  x4n+a:2,,?/2n+2/-,n  by  x2n+xy,-f-2/2n. 

(5)  Divide  am+n6n  —  4am+n-1&2n  —  27ara+n-2i3n  +  42am+n-36<n  by  an6» 
— 7a1>-1b'2n. 

(6)  Divide  a3ra-2"62Pc— a2m+n-161_'>cn  +  a-n&-Iem  +  a3m-nj3p+2cn  _  a2ro+2n-i&3 

c8n-l_j_Jp+lcm+n-l  {jy   Q-n^-p-l^.  fee"-1. 


ANSWERS. 


(2)  x+y. 

(3)  (?n-1+an-Ba;+aH-«a^+ 


an-3x3— xn 


c — x 

(4)  a;2n_;rn2/n_|_^2n> 

(5)  am+3am-lbn—6am-2b°n. 

(6)  a3m_ni3D+1c— a2ra+2n-1i2cn  +  6Pcm. 

EXAMPLES  WITH   LITERAL   COEFFICIENTS.* 

(1)  Divide  ax54-ax4+ix44-ax34-6x3+c.r3+ax2+6x2+cx2+6x+cx4-c  by 
ax^-^-bx-^-c. 

Arrange  the  terms  of  the  dividend  in  the  following  manner,  in  order  to  keep 
the  operation  within  the  breadth  of  the  page. 


ax2+fcx-J-c)  ax5-}- a 
b 


xA-\-a 
b 


Xs  4-  a 
b 


x2+?; 
c 


ax5 +6  x4-f-c  x3 


x-f-c  (x3-f•x24-•r+l■ 


a  x4-f-fl 
6 


b 


xJ 


ax*-\-b  xs+cx3 


a  x3-{-a 
b 

x2-f  Z> 
c 

X 

a  x3-\-b  x2-\-c  x 

a  x3-f-6  x-\-c 

a  x2-f-6  x-\-c 

*  The  literal  multipliers  of  each  power  of  the  letter  of  arrangement  are  to  be  collected 
together,  and  regarded  as  a  polynomial  coefficient  of  that  power,  which  is  to  be  treated 
exactly  in  the  process  of  division  as  a  numerical  coefficient  would  be,  observing  only  the 
four  ground  rules  applicable  to  polynomials  instead  of  numbers. 


28 


ALGEBRA. 


462 


Divid. 


Product 
to  sub- 
tract. 


1st  rem. 
or  2d 
divid. 


Product 
to  sub- « 
tract. 


2d  rem. 
or  3d 
divid. 

Product 
to  sub- 
tract. 

3d  rem. 


a3— 1663 


+ 
+ 


+ 


+ 


462c 
26c2 
2c3 

863 

462c 

26c2 


<z2+3264 

—  863c 

—  A¥c 
+  26c3 

—  c4 


(2)* 

a— 32b5 
+  1664c 
+   863c2 
—  462c3 


Divis. 


—  863 

—  c3 


a2+3264 

—  863c 

—  462c: 
+  26c3 

—  c* 

—  166* 

863c 

462c2 

462c2 

26c3 

c4 


a— 32b5 
+  1664c 
+  863c2 
—  46V 


+ 
+ 

+ 


+  1664 
—  462c2 


a— 32b5 
+  1664c 
+   863c2 

—  46sc3 
+  3265 

—  1664c 

—  863e2 
+  462c-3 


o. 


1st  Partial  Division. 
4b2—  c3   {  26  +  c 
-c 


•  c2   (  2b-\-c 

•  26c  \  26  -( 


-26c— c2 
+  c2 


2<i  Partial  Division. 
—8b3—  c3    f  26  +  c 


+  462c   (  — 462+26c— <* 


+  462c—  c3 
—26c2 


■  26c2—  c3 
+   c3 


3d  Partial  Division. 
64_ 462c2  (  26+c 

46*7 


l— 462c2  <  26+c 
— 863c   (  8b3— < 


■863c— 462c2 
+  462c2 


(3)  Divide  x3+ax2+6x+c  by  x — r. 

x—r)  x3+ax2+6x+c  (x2+(r+a)x+(r2+ar+6) 
x3 — rx2 

(r+a)x2+6x 
(r+a)x2 — (^-{•arfx 


(r2+ar+6)x+c 
(r2+ar+6)x— (rs+a^+ir) 

r3+ar2+6r+c,  remainder 

In  the  preceding  and  similar  examples,  the  remainder  differs  only  from  the 
dividend  in  having  r  instead  of  x.  That  this  is  always  the  case  when  the 
iivisor  is  x  minus  some  quantity,  will  be  shown  hereafter.  (Art.  238,  Pr.  I.* 

(4)  Divide  x3 — <zx2+6x — c  by  x — r. 

(5)  Divide  x3  —  (a  +  6+c)x2+(a6  +  6c+ca)x — a6c  by  x — a. 

(6)  Divide  x3— (a+2).t'2+(2a  +  6)x— 26  by  x— 2. 

(7)  Divide  lla26— 19a6c+10a3— 15a2c+3a62+156c2— 562c  by  5a2+3ai» 
— 56c. 

(8)  Divide  x3— (a+6  +  d)x2+(ad+6rf+c)x— cd  by  x2— (a+6)x+c. 

(9)  Divide  xm+j?xm-1+gxm-2+rxm-3+,  &c...  .  +  te+«  by  x— a. 

*  N.B.  The  signs  of  the  products  to  subtract  are  actually  changed  in  this  example  before 
they  are  written;  a  method  which  is  sometimes  practised.  Their  first  terms  need  not  bo 
written,  since  they  aro  cancelled  by  the  first  terms  of  the  corresponding  dividends. 


DIVISION. 


29 


(10)  Divide 


a4 

x^+a5 

x3 — aBb 

x24-a4&3 

—a*b 

—a4b 

—2a4b2 

+  2a3Z>4 

+a262 

+  a362 

—  ab5 

— ab* 

-\-a"b3 

-a2£6  by 


a* 


— a& 


x2+a3x— a263. 


When  there  are  negative  exponents  of  the  letter  of  arrangement,  they 
come  after  the  term  containing  x",  i.  e.,  the  term  in  which  x  does  not  appear, 
those  which  have  the  greatest  absolute  value  being  placed  last. 

(11)  Divide  — x3— x2-}-10x+§—  '/aT*— ^ar'-f  ftr-9  by  x2— 2x— 2+1*-* 
+|x-». 


ANSWERS. 

(4)  x2+(r— <z)x+(r2- 

-ar-^-b),  and  remainder 

is  r3 — a?-2-{-£r — c. 

(5)  x2— \b-\-c)x+bc. 

(6)  x2— ax+6. 

(7)  2a+6  — 3c. 

(8)  x—d. 

(9)  x^-f-a 

in-B4-  a2 
+  ? 

xm-3_|_a3 

-fa2?? 

+ aq 

xm-4-f-,  &c. 

4-ara-2p 

-\-am~3q 
_j_  am— ir- 

(10)       a2 

+  62 

x2— a26 
— a&2 

x+ 

V. 

+'t 

(11)  -x- 

-3 

+  2.r 

-i_ 

21.  In  those  cases  in  which  the  division  does  not  terminate,  and  the  quotient 
may  be  continued  to  an  unlimited  number  of  terms,  the  quotient  is  termed  an 
infinite  teries,  and  then  the  successive  terms  of  the  quotient  are  generally  reg- 
ulated bv  a  law  which,  in  most  cases,  is  readily  discoverable. 


EXAMPLES. 


(1) 

Divide  1 

by  1 — x. 

1—  x) 

1— X 

+x 
4-x— X2 

-fx2 

-j-X2  —  X3 

-fx3 

The  quotient  in  this  case  is  called  an  infinite  series,  and  the  law  of  formation 
of  this  series  is,  that  any  term  in  the  quotient  is  the  product  of  the  immedi- 
ately preceding  term  by  x. 

(2)  Divide  1  by  1-f  x.  Ans.  1— x-f x2— x'-fx4— ... 

(3)  Divide  1-f  x  by  1— x.  Ans.  1-f  2r+2.t--f  2.^+ 2x*+ .. 


Ans.  xr1 — x~2-f-x- 


•-fz-s — 


(4)  Divide  1  by  x-f  1. 

(5)  Divide  x — a  by  x — b. 

Ans.  1  — (a  —  b)x-x  —  (a— b)bx~"— (a— o^x-3 — 

(6)  Divide  1  by  1—  2x-f  x2.  Ans.  1-f  2x-f  3x2-f  4x3+ 5x*-f .... 


30  ALGEBRA. 

22.  When  a  polynomial  is  the  product  of  two  or  more  factors,  it  is  often 
requisite  to  resolve  it  into  the  factors  of  which  it  is  composed,  and  merely  to 
indicate  the  multiplication.  This  can  frequently  be  done  by  inspection,  and 
ly  the  aid  of  the  following  formulas  : 

(x+a){x+b)=x"-+{a  +  b)x-\-ab (1) 

{x+a)(x  —  b)=x*+  (a  —  b)x—ab (2) 

(x— a)(x+6)=x2  —  {a  —  b)x— ab (3) 

(x—a){x—b)=x°— (a+b)x+ab (4) 

(a  +  b){a  —  b)=a'i—V (5) 

(n+l)(n+l)=»a+2n+l (6) 

(n— J)(n— l)=ns— 2n-f  1 (7) 

EXAMPLES. 

(1)  Resolve  ax'2-\-bx'2 — ex2  into  its  component  factors. 

Here  ax2+bx3— cx2=x2(a  +  fc  — c). 

(2)  Transform  the  expression  n?-\-2n?-trn  into  factors. 

Here  n?-\-2rii-\-n=n{n2-\-2n+l) 

=n(n+l)(»+l)by  (6) 
=n(n-\-l)2. 

(3)  Decompose  the  expression  x2— x— 72  into  two  factors. 

By  inspecting  formula  (3),  we  have  — 1  =  —  9+8,  and  —72=— 9X8 
hence  x2— x— 72=(x— 9)(x+8). 

(4)  Decompose  5a2bc-\-lQab-c-{-l5abc2  into  two  factors. 

(5)  Transform  3m4n6— 6m3n5p-\-3m2n4j)z  into  factors. 

(6)  Transform  3fe3c — 35c3  into  factors. 

(7)  Decompose  x2-r-8x-fl5  into  two  factors. 

(8)  Decompose  x3 — 2x2 — 15x  into  three  factors. 

(9)  Decompose  x2— x— 30  into  factors. 

(10)  Transform  a2 — 62-f-26c — c2  into  two  factors. 

(11)  Transform  a2x— x3  into  factors. 


ANSWERS. 


(4)  5abc(a+2b-\-3c). 

(5)  3m2n4(mn  — pf. 

(6)  3bc{b  +  c){b  — c). 

(7)  (x+3)(x+5). 


(8)  x(x+3)(x— 5). 

(9)  (.r+5)(x-6). 

(10)  (a  +  b— c){a  —  b+c). 

(11)  x(a-\-x)(a — x). 


23.  By  the  usual  process  of  division  we  might  obtain  the  quotient  of  a" — 6" 
divided  by  a — b,  when  any  particular  number  is  substituted  for  n;  but  we 
shall  here  prove  generally  that  an—bn  is  always  exactly  divisible  by  a— b,  and 
exhibit  the  quotient. 

Tt  is  required  to  divide  an— 6n  by  a  —  b. 

a-b)a"-b»       (a"->  + ^ -' 

an  —  an~xb 
Rem.         a"-ifc_Z>n; 
Rem.  under  another  form,  b(an-1   —  &n_1). 

Hence,  ^T^  +        a-0  (1> 


DIVISION.  31 

Now  it  appears  from  this  result,  that  a"  —  b"  will  be  exactly  divisible  by 
a — b,  if  «n_i —  b""1  be  divisible  by  a  —  b  ;  that  is,  if  the  difference  of  the  same 
powers  of  two  quantities  is  divisible  by  their  difference,  then  the  difference 
of  the  powers  of  the  next  higher  degree  is  also  divisible  by  that  difference. 

But  a- — b2  is  exactly  divisible  by  a — i,  and  we  have 

^=b=a  +  b <2> 

And  since  a2 — Z>2  is  divisible  by  a  —  b,  it  appears,  from  what  has  been  just 
proved,  that  a3 — b3  must  be  exactly  divisible  by  a — b  ;  and  since  a3 — b3  is  di- 
visible, a4 — b4  must  be  divisible,  and  so  on  ad  infinitum. 

Hence,  generally,  an — bn  will  always  be  exactly  divisible  by  a  —  b,  and  give 
the  quotient 

an  —  bn 

— — T-=a'w+a,1-s&  +  an-3&2+ a'bn'3-\-nbn~'^.ba'1 (5) 

In  a  similar  manner,  we  find,  when  n  is  an  odd  number, 

a"4-bn 

— ~-r=an-1  —  an-26  +  an-362— Ma-ba~3— abn~3+bn~1   ....   (6) 

a-f-o  i  i  i  \ 

And  when  n  is  an  even  number 

an bn 

— — =a"-i— a°-26+an-3i2— _a2in-3+aZ>n-2  —  bn~l   ....  (7) 

a-\-b 

By  substituting  particular  numbers  for  n,  in  the  formulas  (5),  (G),  (7),  we 
may  deduce  various  algebraical  formulas,  several  of  which  will  be  found  in  the 
following  deductions  from  the  rules  of  multiplication  and  division. 

USEFUL    ALGEBRAIC   FORMULAS. 

(1)  a3— ¥=(a+b){a  —  b). 

(2)  a4  — 6«=(as+&s)(a3— i2)  =  (a2+62)(a+6)(a  —  b). 

(3)  a3— 63=(a2+a&-f-i2)(a  —  b). 

(4)  a*+b°={a*— ab+b°~)(a+b). 

(5)  a«— bs=(a3+b3){a3  —  b3)  =  {a3+b3){ai+ab  +  b*)(a  —  b). 

(6)  a6— b«={a3+b3)(a3— Z>3)  =  (a3— 63)(a3— ab  +  b*){a+b). 

(7)  ae— &«=(a3+63)(a3— &3)  =  (<z2— 62)(a4+a2&2+&4). 

(8)  a6— i«=(a  +  6)(a— &)(a2+a&-f-62)(«2— a&+62). 

(9)  (a"-—b-)^-{a—b)=a-{-b. 

(10)  (a3— 6s)4-(«— &)=a2-l-a&+&*. 

(11)  (a3+63)-i-(a  +  6)=aa— a&+Z>2. 

(12)  (a4— &4)^-(a+&)=a3— a2&-fa&2— &3. 

(13)  (a5— &*)-i-(a  — &)=a4+a3Z>  +  a2&2+a&3-|-Z>4. 

(14)  (a6+65)-H(a  +  ^)=»«4— a3i+a262— a&3-f  &4- 

(15)  (a6— i6)-^-(a2— 62)=a4-fa':i-+64. 

DIVISION  BY  DETACHED  COEFFICIENTS. 

24.  Arrange  the  terms  of  the  divisor  and  dividend  according  to  the  success- 
ive powers  of  the  letter,  or  letters,  common  to  both ;  write  down  simply  the 
coefficients  with  their  respective  signs,  supplying  the  coefficients  of  the  absent 
terms  with  zeros,  and  proceed  as  usual.  Divide  the  highest  power  of  the 
omitted  letters  in  the  dividend  by  that  of  the  omitted  letters  in  the  divisor, 
and  the  result  will  be  the  literal  part  of  the  first  term  in  the  quotient.     The 


32  ALGEBRA. 

literal  parts  of  the  successive  terms  follow  the  same  law  of  increase  or  de- 
crease as  those  in  the  dividend.  The  coefficients  prefixed  to  the  literal  parts 
will  give  the  complete  quotient,  omitting  those  terms  whose  coefficients  are 
Eero. 

EXAMPLES. 

(i)  Divide  6a4— 96  by  3a— 6. 

3_6)  6+   0+0+0—96  (2+4  +  8+16 
6—12 


12 

12- 

-24 

24 

24- 

-48 

48- 

-96 

48- 

-96 

But  a^-^-a^a3,  and  the  literal  parts  of  the  successive  terms,  are,  therefore 
a3,  a3,  a1,  a0,  or  a3,  a3,  a,  1 ;  hence,  2a3+4a2+8a+16=  quotient. 

(2)  Divide  8a5— 4a4ar— 2a3x3+a2x3  by  4a3— x3. 

4  +  0—1)  8  —  4—2  +  1  (2—1 
8+0  —  2 
_4  +  0+l 
_4_0  +  l 


Now,  ab-±-a?=.a?;  hence  a3  and  cfiz  are  the  literal  parts  of  the  terms  in  the 
quotient,  for  there  are  only  two  coefficients  in  the  quotient ;  therefore 

2a3 — a3x=  quotient  required. 

(3)  Divided— 3az3— 8a3x3+18a3a:— 8a4  by  a:3+2a:r— 2a3. 

(4)  Divide  32/3+3x2/3— 4x*y— 4Z3  by  x-\-y. 

(5)  Divide  10a4— 27a3x+34a2x2— 18ax*— 8X4  by  2a3— 3ax+4:r*. 

(6)  Divide  a4+5a3+a+5  by  a3+l. 


(3)  x2— 5aa:+4a2. 

(4)  _4x2+3i/3. 


ANSWERS. 

(5)  5a2— 6aa:— 2x*. 

(6)  a +5. 


SYNTHETIC  DIVISION. 

RULE.* 

25.  (1)  Divide  the  divisor  and  dividend  by  the  coefficient  of  the  first  term  in 

*  The  rule  here  given  for  Synthetic  Division  is  due  to  the  late  W.  G.  Horner,  Esq.,  of 
Bath,  whose  researches  in  science  have  issued  in  several  elegant  and  useful  processes, 
especially  in  the  higher  branches  of  algebra,  and  in  the  evolution  of  the  roots  of  equation  of  all 
dimensions. 

In  the  common  method  of  division,  the  several  terms  in  the  divisor  are  multiplied  by  the 
first  term  in  the  quotient,  and  the  product  subtracted  from  the  dividend ;  but  subtraction  is 
performed  by  changing  all  the  signs  of  the  quantities  to  be  subtracted,  and  then  addir.^ 
the  several  tenns  in  the  lower  line  to  the  similar  terms  in  the  higher.  If,  therefore,  the 
signs  of  the  tenns  in  the  divisor  were  changed,  we  should  have  to  add  the  product  of  the 
divisor  and  quotient  instead  of  subtracting  it.  By  this  process,  then,  the  second  dividend 
would  be  identically  the  same  as  by  the  usual  method.  We  may  omit  altogether  the 
products  of  the  first  term  in  the  divisor  by  the  successive  tenns  in  the  quotient,  because 
in  the  usual  method  the  first  term  in  each  successive  dividend  is  cancelled  by  these  prod- 
acts.    Omitting,  therefore,  these  products,  the  coefficient  of  the  first  term  in  any  dividend 


DIVISION.  33 

the  divisor,  which  will  make  the  leading  coefficient  of  the  divisor  'ju.ity,  and 
the  first  term  of  the  quotient  will  be  identical  with  that  of  the  dividend. 

(2)  Set  the  coefficients  of  the  dividend  in  a  horizontal  line  with  their  proper 
signs,  and  those  of  the  divisor,  with  the  signs  all  changed  except  that  of  the 
first,  in  a  vertical  column  on  the  right  or  left,  drawing  a  line  under  the  whole, 
underneath  which  to  write  the  quotient. 

(3)  Multiply  all  the  terms  so  changed  by  tho  first  term  in  the  quotient,  and 
place  the  products  successively  under  the  corresponding  terms  of  the  dividend, 
in  a  diagonal  column. 

(4)  Add  the  results  in  the  second  column,  which  will  give  the  second  term 
of  the  quotient ;  and  multiply  the  changed  terms  in  the  divisor  by  this,  placing 
the  products  in  a  diagonal  series,  as  before. 

(5)  Add  the  results  in  the  third  column  for  the  next  term  in  the  quotient, 
by  which,  again,  multiply  the  changed  terms  in  the  divisor,  placing  the  prod- 
ucts as  before. 

(G)  This  process,  continued  till  the  last  line  of  products  extends  as  far  to  the 
right  as  the  dividend,  will  give  the  same  series  of  terms  as  the  usual  mode  of 
division. 

EXAMPLES. 

(1)  Divide  a5— Sa'x+lOa3^— lOaW+Sax*— .r5  by  a2— 2ax+x-. 


1 

1—5+10— 10+5  — 1 

+  2 

+  o_  6_|_   6—2 

— 1 

—  1+   3  —  3  +  1 

1  —  3  +   3_   l     *     * 
Hence  a3 — 3a2.r+3ax2 — x3=  quotient. 

In  this  example  the  coefficients  of  the  dividend  are  written  horizontally,  and 
those  of  the  divisor  vertically,  with  all  tho  signs  of  the  latter  changed,  except 
the  first.  Then  +2  and  — 1,  the  changed  terms  in  the  divisor,  are  multiplied 
by  1,  the  first  term  of  the  quotient,  which  is  written  in  the  horizontal  line  at 
the  bottom,  and  is  the  same  as  the  first  term  of  the  dividend ;  the  products 
+  2  and  — 1  are  placed  diagonally,  under  — 5  and  +10,  the  corresponding 
lerms  of  the  dividend.  Then,  by  adding  the  second  column,  we  have  — 3  for 
the  second  term  in  the  quotient,  and  the  changed  terms  +2  and  — 1  in  the 
divisor,  multiplied  by  — 3,  give  — 6  and  +3,  which  are  placed  diagonally  un- 
der +  10  and  — 10.  The  sum  of  the  third  column  is  +3,  the  next  term  in 
the  quotient,  which,  multiplied  into  the  changed  terms  of  the  divisor,  gives 
+  G — 3,  for  the  next  diagonal  column.  The  sum  of  the  fourth  column  is  — 1, 
and  by  this  we  obtain  the  last  diagonal  column  — 2+1.  The  process  here 
terminates,  and  the  sums  of  the  fifth  and  sixth  columns  are  zero,  which  shows 
that  there  is  no  remainder.  If  the  last  terms  did  not  reduce  to  zero  by  addi- 
tion, their  sum  would  be  the  coefficients  of  the  remainder ;  the  quotient  is  com- 
pleted by  restoring  the  letters,  as  in  detached  coefficients. 

Having  made  the  coefficient  of  the  first  term  in  the  divisor  unity,  that  co- 
will  be  the  coefficient  of  the  succeeding  term  in  the  quotient,  the  coefficient  in  the  first 
term  of  the  divisor  being  unity ;  for  in  all  cases  it  can  be  made  unity  by  dividing:  both 
divisor  and  dividend  by  the  coefficient  of  the  first  term  in  the  divisor.  The  operation,  tbus 
simplified,  may,  however,  be  farther  abridged  by  omitting  the  successive  additions,  except 
so  much  only  as  is  necessary  to  show  the  first  term  in  each  dividend,  which,  as  before  re 
marked,  is  also  the  coefficient  of  the  succeeding  term  in  the  quotient. 

c 


34  ALGEBRA. 

efficient  may  be  omitted  entirely,  since  it  is  of  no  use  whatever  in  continuin 
the  operation  here  described. 

(2)  Divide  x*— hx*+\5x*— 24x3+27x2— 13x+5  by  z4— 2rJ+4a;8— 2x+l 
l_5_j_15_24  +  27  — 13+5 
+2    _|_o_  6-1-10 
—  4  —  4  +  12—20 

+  2  +   2—   6+10 

—1  —   1+   3—5 


1_3+   5       0        0        0      0 
Hence  .r2 — 3:r+5=z:  quotient  required. 

(3)  Divide  a5+2a4b+3aW— a°~b3— 2ab*— 365  by  a2+2a&+3&». 
1  +  2+3—1—2—3 
—2+0  +  0  +  2 
—3+0+0+3 


—2 
—3 


1+0  +  0—1 
Hence  a3+0-a2Z>+0-aZ>2  —  b3=a3— &3=  quotient. 

(4)  Divide  1— x  by  1+z.  Ans.  1— 2x+2:e2— 2r3+,  &c. 

(5)  Divide  1  by  1—x.  Ans.  l+.r+^+x3-}-.  &c. 

(6)  Divide  x1 — y7  by  x — y.     Ans.  x&-\-x6y-{-x4y'i-\-xzyz-irx^yi-\-xy'a-\-yt. 

(7)  Divide  a6— 3a4.r2+3a2:r4— a*  by  a3— 3a2.r+3az2  — r5. 

Ans.  a3+3a2x+3ax2+x3. 

(8)  Divide  a5— 5a4ar+10a3:c2— IQcPtf+bax*— x5  by  a2— 2ax\-x". 

Ans.  a3— 3a2a;+3aa;2— 2°. 

(9)  Divide  4?/6— 24t/5+60?/4— 802/3+60?/2— 24^+4  by  2i/2— 4?/+2. 

Ans.  2i/4— 8i/3+12i/2— 83/+2 

THE  GREATEST  COMMON  MEASURE. 

26.  A  measure  of  a  quantity  is  any  quantity  that  is  contained  in  it  exactly, 
or  divides  it  without  a  remainder ;  and,  on  the  other  hand,  a  multiple  of  a 
quantity  is  any  quantity  that  contains  it  exactly.     Thus,  5  is  a  measure  of  15, 
and  15  is  a  multiple  of  5  ;  for  5  is  contained  in  15  exactly  3  times,  and  15  con 
tains  5  exactly  3  times,  or  is  produced  by  multiplying  5. 

27.  A  common  measure,  or  common  divisor,  of  two  or  more  quantities,  is  a 
quantity  which  is  contained  exactly  in  each  of  them. 

28.  The  greatest  common  measure,  of  two  or  more  quantities,  is  composed 
of  all  the  prime*  factors,  whether  numerical,  monomial,  or  polynomial  factors, 
«ommon  to  each  of  the  quantities ;  3x  is  a  common  measure  of  12ax  and 
I8bx,  and  6x  is  the  greatest  common  measure  of  12a.r  and  18bx.  The  great- 
est common  divisor  of  2x7a(b-\-c)d  and  2x3a?n(b-{-c)  is  composed  of  the 
common  prime  factors  2a(b-\-c) ;  the  factors  id  of  the  one  and  3  of  the  other 
make  no  part  of  the  common  divisor. 

29.   To  find  the  greatest  common  measure  of  two  polynomials. 

Arrange  the  polynomials  according  to  the  powers  of  some  letter,  and  divide 

that  which  contains  the  highest  power  of  the  letter  by  the  other,  as  in  division; 

then  divide  the  last  divisor  by  the  remainder  arising  from  the  first  division  ; 

consider  the  remainder  that  arises  from  this  second  division  as  a  divisor,  and 

*  A  prime  number  or  a  prime  algebraic  quantity  is  one  whicb  is  divisible  only  by  itself 
or  unity. 


GREATEST  COMMON  MEASURE.  35 

the  last  divisor  as  tho  corresponding  dividend,  and  continue  this  process  of  di- 
vision till  the  remainder  is  0  ;  then  the  last  divisor  is  the  greatest  common 
measure. 

Note  1.  When  the  highest  power  of  tho  leading  quantity  is  the  same  in 
both  polynomials,  it  is  indifferent  which  of  the  polynomials  is  made  the  divisor, 
the  only  guide  being  the  coefficients  of  the  leading  terms  of  the  polynomials. 

Note  2.  If  the  two  given  polynomials  have  a  monomial  factor  common  to  all 
the  terms  of  both,  it  may  be  suppressed  ;  but  as  it  forms  part  of  the  common 
measure  (28),  it  must  be  restored  at  the  end  of  the  process  by  multiplying  it 
into  the  common  measure  which  is  in  consequence  obtained. 

Note  3.  If  any  divisor  contain  a  factor,  which  is  not  a  factor  also  of  the  divi 
dend,  that  factor  may  be  rejected,  as  such  factor  can  form  no  part  of  the  great- 
est common  measure,  which  is  composed  of  the  common  factors  alone. 

Note  4.  If  the  coefficient  of  the  leading  term  of  any  dividend  be  not  divisible 
by  that  of  the  divisor,  it  may  be  rendered  so  by  multiplying  every  term  of  the 
dividend  by  a  proper  factor,  to  make  it  divisible.  This  new  factor  thus  jntro 
duced,  not  being  a  common  factor,  does  not  affect  the  common  measure. 

If  it  were  already  a  factor  of  the  divisor,  it  could  not  be  thus  used  ;  the 
remedy,  in  this  case,  would  be  to  suppress  it  in  the  divisor,  according  to  Note  3. 

In  order  to  prove  the  truth  of  this  rule,  we  shall  premise  two  lemmas.* 

Lemma  1.  If  a  quantity  measure  another  quantity,  it  will  also  measure 
any  multiple  of  that  quantity.  Thus,  if  d  measures  a,  it  will  also  measure  m 
times  a,  or  ma;  for,  let  a=hd,  then  ma=mhd,  and,  therefore,  d  measures 
ma,  the  quotient  being  mil. 

Lemma  2.  If  a  quantity  measure  two  other  quantities,  it  will  also  measure 
both  their  sum  and  difference,  or  any  multiples  of  them.  For,  let  a=hd,  and 
b=kd,  then  d  measures  both  a  and  b;  hence  a±b=hd^z7cd=d(h-^zJc), 
and,  therefore,  d  measures  both  a-\-b  and  a — b,  the  quotient  being  h-\-k  in 
the  foi-mer  case,  and  h — lc  in  the  latter :  and  by  lemma  1,  d  measures  any 
multiples  of  a-\-b  and  a — b. 

Now,  let  a  and  b  be  two  polynomials,  or  the  terms  of  a  fraction,  and  let 

a  divided  by  b  leave  a  remainder  c 

b c d  b)  a  (m 

c d  leave  no  remainder,  as  is  shown  m  b 

in  the  marginal  scheme.     Then  we  have,  by  the  c)  b  (n 

nature  of  division,  these  six  equalities,  viz. :  n  c 

a—mb=.c  ....  (1)       a=mb-\-c   ....  (4)  d)  c  (p 

b — nc  =d  ....  (2)       b=nc-\-d  ....  (5)  p  d 

c—pd=0  ....  (3)       c=pd         ....  (6) 
where  the  equalities  marked  (4),  (5),  (G)  aro  not  deduced  from  those  marked 
(1),  (2),  (3),  but  from  the  consideration  that  the  dividend  is  always  equal  to 
the  product  of  the  divisor  and  quotient,  increased  by  the  remainder. 

Now,  by  (6)  it  is  obvious  that  d  measures  c,  since  c=pd ;  hence  (Lemma 
1)  d  measures  nc,  and  it  likewise  measures  itself;  therefore  (Lemma  2)  d 
measures  nc-\-d,  which  by  (5)  is  equal  to  b  ;  hence,  again,  d,  measuring  b  and 
c,  measures  mb-\-c  by  the  Lemmas  1  and  2. 

*  A  lemma  is  a  preparatory  proposition,  to  aid  in  the  demonstration  of  the  main  prooosi. 
tion  which  follows  it. 


36  ALGEBRA. 

.•.  d  measures  a,  which  is  equal  to  mb-\-c  by  (4). 

Hence  d  measures  both  the  polynomials  a  and  b,  and  is  consequently  a 
common  measure  of  these  polynomials ;  but  d  is  also  the  greatest  common 
measure  of  a  and  b  ;  for  if  d'  is  a  greater  common  measure  of  a  and  b  than  d 
is,  it  is  obvious  that  by  (1)  d'  measures  a — mb,  or  c  ;  and  d'  measuring  both  b 
and  c,  it  measures  b — nc,  or  d  by  (2) ;  hence  d'  measures  d,  which  is  absurd, 
since  no  quantity  measures  a  quantity  less  than  itself;  therefore  d  is  the 
greatest  common  measure.  Q.  E.  D.* 

30.  If  the  greatest  common  measure  of  three  quantities  be  required,  find 
the  greatest  common  measure  of  two  of  them,  and  then  that  of  this  measure 
and  the  remaining  quantity  will  be  the  greatest  common  measure  of  all  three. f 

31.  If  the  two  polynomials  be  the  terms  of  a  fraction,  as  r»  and  d  their 

greatest  common  measure,  then  we  may  put  a=da',  and  b=db' ;   hence 

-  =  —  =— ,  and,  since  a',  b'  contain  no  common  factor  (28),  by  dividing  both 
b     db'      b' 

numerator  and  denominator  of  a  fraction  by  their  greatest  common  measure, 
the  resulting  fraction  will  be  simplified  to  its  utmost  extent,  and  thus  the  pro- 
posed fraction  will  be  reduced  to  its  lowest  terms. 

*  These  letters  stand  for  the  Latin  words  quod  erat  demonstrandum,  signifying  which 
was  to  be  demonstrated.  Another  mode  of  demonstrating  the  same  is  as  follows :  Let  A 
and  B  represent  the  two  given  quanties,  D  their  greatest  common  divisor,  Q,  the  quotient 
of  A  by  B,  and  R  the  remainder.  We  shall  first  prove  that  the  greatest  common  divisor 
of  A  and  B  is  the  same  as  the  greatest  common  divisor  of  B  and  R.  Represent  the  latter 
byD'. 

a      ^  i  w        A     Baj  R       a  A     Ba.  R 
A=Ba+R,  •••  5=^+5.  ^d  ^=W+W- 

A  and  B  being  divisible  by  D,  B  must  be,  because  a  whole  number  can  not  be  equal  to 
a  whole  number  plus  a  fraction.  Again,  B  and  R  being  divisible  by  D',  A  must  be,  for  the 
sum  of  two  whole  numbers  can  not  equal  a  fraction.  Finally,  D,  a  common  divisor  of  B 
and  R,  can  not  be  greater  than  their  greatest  common  divisor  D' ;  and  D',  a  c .  d .  of  A  and 
B,  can  not  be  greater  than  their  g .  c .  d .  D ;  i.e.,  D  can  not  be  greater  than  D',  and  D'  can 
not  be  greater  than  D. 

Or  thus :  since 

A=BGl+R, 
the  greatest  common  divisor  D  of  A  and  B,  must  divide  R.    Represent  the  three  quotients 
by  A',  B',  andR';  then 

A'=B'Q,+R'. 
B'  and  R'  have  no  farther  common  factor,  for  if  they  had,  it  must  by  this  equality  divide 
A;  then  A'  and  B'  would  have  still  a  common  factor,  and  D,  the  greatest  common  divisor 
of  A  and  B,  would  not  contain  all  the  common  factors  of  these  quantities,  which  is  contrary 
to  the  definition.  Since  B'  and  R',  which  are  the  quotients  of  B  and  R  by  D,  can  have  no 
farther  common  factor,  it  follows  that  the  greatest  common  divisor  of  B  and  R  is  equal  to 
D  ;  then  it  is  the  same,  as  that  of  the  quantities  A  and  B. 

In  pursuing  the  rule  for  finding  the  g .c.d.,  we  arrive  at  a  remainder  which  exactly  di- 
vides the  preceding  divisor,  and  which  is,  therefore,  the  g.c.d.  of  itself  and  this  preced- 
ing divisor ;  also  by  the  above  demonstration  of  that  divisor  and  its  dividend,  and  so  on  up 
to  the  given  quantities. 

t  For  suppose  we  have  the  three  quantities  A,  B,  C;  let  D  be  the  greatest  common  di- 
visor of  A  and  B,  and  D'  that  of  D  and  C.  According  to  the  definition,  D  is  the  product  of 
the  common  factors  of  A  and  B,  and  D'  is  that  of  the  common  factors  of  D  and  C  ;  then  D'  is 
the  product  of  the  common  factors  of  the  three  quantities  A,  B,  C ;  therefora  D'  is  their 
greatest  common  divisor. 


GREATEST  COMMON  MEASURE.  37 

EXAMPLES. 

(1)  What  is  the  greatest  common  measure  of  Ax^y^z*  and  8x*y3z'2l 

Here  4  is  the  greatest  common  measure  of  4  and  8,  and  x2y3z2  is  that  of  the 
literal  parts;  hence  4x2y3z2  is  the  greatest  common  measure  required. 

(2)  Find  the  greatest  common  measure  of     ^_    . 

%J 

Xs — xy2 

xy2-\-y3=y2(x-\-y) ;  rejecting  the  factor  y- 
x-Ly)     x"—y2  (x—y 
x2-\-xy 
—xy  —  y" 
— xy — y2. 

Hence  x-\-y  is  the  greatest  common  measure  sought,  and 
a?  i  ,,3     (xs_|_j,3w  [x+y)     x2—xy-\-y2 

n  J  ==) — 1-^-+ — ) — L-^= =  reduced  fraction. 

a8— f     (a-1'3— 2/2)-r(^+2/)  *— 9 

(3)  Required  the  greatest  common  measure  of  the  two  polynomials 

6a3—  6a2y+2ay2—2y3  ....  (a) 
12a2— lhay  +3y*  ....  (&)• 

Here  6a3—  6a2y+2aif— 22/3=2(3a3— 3a2y+ay2— y3) 

12a2—15ay  +3?/3  =3(4a2— 5a?/  +r) ; 

And  therefore,  by  suppressing  the  factors  2  and  3,  which  have  no  common 
measure,  and  which,  not  being  common  factors  of  the  two  given  quantities,  do 
not  affect  the  common  divisor,  we  have  to  find  the  greatest  common  measure 
of 

3a3— 3a2y+ay2— y3  and  4a2—5ay-\-y2. 
4a2— 5ay+y~)  3a3—  3a?y  +  ay"—  y3 

_4 

12a3— 12a2^+4a?/2—  4y3  (3a 

12a3— 15a2?/ -j- 3a?/2 

3a"y  +  ay2 —  4y3 
4 


12a2?/ +  4a?/2— 16y:s  {3y 
I2a2y  —  I5ay2+  3y3 


19ay2—19y3=l9y2  (  a— y) 
Suppressing  19y2,  by  note  3,  rule, 

a—y)  4a"—5ay+y"  (4a— y 
4a2 — 4ay 

—  ay+y2 


Hence  a  — y  is  the  greatest  common  measure  of  the  polynomials  a  and  b. 

The  factor  4  is  introduced  into  the  divkleud  in  this  example  to  render  it  di- 
visible by  the  divisor.  This  can  be  done,  because  4  is  not  a  factor  of  every 
term  of  the  divisor,  and  therefore  not  a  factor  of  the  divisor.  The  quantities 
employed,  after  introducing  or  suppressing  factors,  are  different  from  the  given, 
but  as  they  have  the  same  greatest  common  divisor,  and  as  the  object  is  to  find 
this,  the  circumstance  is  immaterial. 

(4)  Required  the  greatest  common  measure  of  the  terms  of  the  fraction 


38 


ALGEBRA. 
aF—ct-x4 


a6  -4-  a3x — a4x2 — aV 
Here  a~  is  a  simple  factor  of  the  numerator,  and  a3  is  a  factor  of  the  denomi- 
nator; hence  a3  is  the  greatest  common  measure  of  these  simple  factors,  which 
must  be  reserved  to  be  introduced  into  the  greatest  common  measure  of  the 
other  factors  of  the  terms  of  the  proposed  fraction  ;  viz. : 
a4 — x4  and  a3+«2^'  — ax2 — x3. 
a34-a3x — ax"1 — x3)  a4 — x4  (a — x 

a4  -f-  a3x — (Px* — ax3 
— d?x  -j-  a2.r3  -|-  ax3 — x4 
— asx — a2.r2  -4-  ax3  -|-  x4 


2a"x2—2x4=2x'i  (a3—  a:2) ;  rejecting  2x3 
a?—x'2)  d?-\-a*x— a.r3— x3  (a-\-x 
a3 — ax2 

a-x — x3 

a"x — x3 


Therefore,  restoring  a2,  the  greatest  common  measure,  is  a2(a2 — .r2). 

gS—cPx4 (as—a?x4)-±a~(a'i—x'2)  cP+x* 

•  a6-\-a5x — a4x3 — a¥     (a6-\-a5x — a4a:3 — a3x3)-^-a\a<1 — x")     a?-\-ax 

ADDITIONAL  EXAMPLES. 

(1)  Find  the  greatest  common  measure  of  2a3a:3,  4a:3?/2,  and  63?y. 

(2)  Find  the  greatest  common  measure  of  the  two  polynomials  a3 — c&b 
+  3aZ>3— 3b3,  and  a2— 5aZ>+4i3. 

(3)  What  is  the  greatest  common  measure  of  x3 — xy*  and  a:2 -j-  2xy -J- y"  1 

(4)  Find  the  greatest  common  measure  of  x8 — ys  and  a43 — y13. 

(5)  Find  the  greatest  common  measure  of  the  polynomials 

(b—c)x^—b(2b— c)a:+&3     (a) 

lb-\-c)x3—b{2b-\-c)x'i+b3x (6). 

(6)  Find  the  greatest  common  measure  of  the  polynomials 

x4—  8r3+21a:3— 20x-f  4 (a) 

2X3— 12a;3 + 21a:  —10  (b). 

(7)  y3—5y2z—4yz*+2z3  and  7;?/2z-f  Kh/z24-3z3. 

(8)  Also  of  (a^+a3a:3+a4)  and  (x4-\-ax3— a3x— a4). 

(9)  Also  of  (7a3— 23a&+6&2)  and  (5a3— 18&a3+llaZ>3— 6b3). 

(10)  Also  of  (5a5+10a4Z>+5a3i2)  and  (a3&  +  2a2&2+2a&3+Z>4). 

(11)  Also  of  (6a5-fl5a4&  — 4a3c2  — 10a26c3)  and  (9a36  —  "27a2Z>c  —  6a&c« 
4-186C3). 

(12)  Also  of  {aa+Y+aybP-\-aabs+bP+l)  and  {aam-\-aan^m-\-b^i). 

(13)  Find  the  g.  c.  d.  of  the  three  quantities  a3-f3a2&  +  3a63+&3,Na2+2a& 
+  62,  and  a2— Z>2. 

ANSWERS. 


(8)  x^+ax+a?. 

(9)  a— 36. 
(10)  a-\-b. 


(11)  3a2— 2c2. 

(12)  aa+Z># 

(13)  a+b. 


(1)  2x2.  (5)  x—b. 

(2)  a— b.  (6)  x— 2. 

(3)  x+y.  (7)  y+z. 

(4)  x—y. 
A  quantity  is  said  to  be  independent  of  a  letter  when  it  does  not  contain  this 

letter,  and,  therefore,  does  not  depend  upon  it  for  its  value. 

Note. — Iu  seeking  for  a  common  divisor,  we  find  ourselves  often  working  with  polynomi- 
als different  from  the  given,  but  always  with  such  as  have  the  same  common  measure  with 
the  given  polynomials. 


GREATEST  COMMON  MEASURE.  3y 

Proposition. — A  divisor  of  a  polynomial,  which  is  independent  of  the  letter 
of  arrangement  of  that  polynomial,  must  divide  separately  each  bf  the  multi- 
pliers of  the  different  powers  of  that  letter. 

Demonstration. — Let  Axm-f-B.rm_1-f-C;rm_2,  &c,  be  the  polynomial,  and 
D  the  divisor.  The  quotient  must  contain  every  power  of  the  letter  of  ar- 
rangement that  the  dividend  does,  since  the  quotient,  multiplied  by  the  divisor, 
must  produce  the  dividend,  and  the  letter  of  arrangement  is  not  contained  in 
the  divisor.  The  quotient  must,  therefore,  be  of  the  form  A'xm-\-B'xm~l 
-\-C'xm~2,  &c,  multiplying  which  by  the  divisor  gives  DA/xm-|-DB'.rm_1 
-f-DC'i"1-2,  &c,  the  original  dividend,  the  multiplier  of  each  power  of  x  in 
which  is  evidently  divisible  by  D.  Q.  E.  D. 

N.B. — A'  is  a  different  quantity  from  A,  B'  from  B,  &c. 

EXAMPLES. 

(1)  Find  a  common  divisor,  independent  of  the  letter  a,  of  the  two  quantities 
b  a-—ca'i-\-b2a—c'2a-\-b"—2bc-\-c'i  and 
b3a*—3b*caa+3bc*a3—c?a3+¥a*—c*a*+b3a—(?a+b3—3bic+3bc'i—ci. 

Collecting  together  in  the  first  of  these  two  quantities  the  multipliers  of  a2  and 
a,  and  observing  that  b2 — 2bc-\-c"  is  the  square  oft — c,  we  hav 

(J_c)a2_|_(&2_c2)a_L.(&_c)2, 

and  from  the  second  by  a  similar  process, 

(b—c)sa*+(b4— c*)a9+(&3— c3)a+(6— c)3. 
The  multipliers  of  the  different  powers  of  a  in  the  two  quantities  are,  there- 
fore, b—c,  b"—c",   (b—c)°,   (6— c)3,  Z>4— c4,  and  b3—c3.     The  only  number 
which  will  divide  them  all  is  their  common  divisor  b — c,  which  is,  therefore 
the  answer  required. 

(2)  Find  the  greatest  common  divisor  of 

(b  —c  )a2— 2b  (b  —c)a-\-b~{b—c)  and 
(b--c-)a-—  &a(&8— <■-•). 
Here  the  common  divisor,  independent  of  a,  is  b — c ;  suppressing  which,  wo 
have  left  the  two  quantities 

a2—  2ba  -\-b-  and 
(o  +c)(a2— Z>2). 
Suppressing  the  factor  (b-\-c)  not  common  to  both,  we  shall  find  the  common 
divisor  of  these  last  two  quantities  to  be  a — 6,  and  the  greatest  common 
divisor  of  the  two  original  quantities  is 

(6 — c)  (a — b)  or  ab — ac — b2-{-bc. 

The  success  of  the  process  for  finding  a  greatest  common  divisor  depends  upon  the  fact 
that  the  quantities  being  arranged  according  to  the  powers  of  some  letter,  each  division 
leads  to  a  remainder  of  a  degree  inferior  to  the  divisor.  When  the  polynomials  contain 
many  terms  of  the  same  degree,  a  precaution  is  necessary,  without  which  this  reduction 
would  not  always  obtain,  which  consists  in  uniting  all  these  terms  under  a  single  multiplier 
Let  there  be  the  two  polynomials : 

A=x3  -\-yx--]-xi  — y-x-\-2yx — y3-\-y'i 
'B=yx°~-\-zfi  -\-y?x+yx  -f  x     -f-y. 
I  write  them  thus :  * 

A=x3+(y+l)x-— (y-— 1y)x— y3  +yz 
B=  (y  +i)x:+(y2+y  +l)x+y. 
The  first  term,  x3,  not  being  divisible  by  (y-\-l)x%  on  account  of  the  factor  y-f-1,  I  know 
(Prop,  above),  that  if  a  quantity  is  arranged  like  the  preceding,  every  divisor  of  this  quantity, 
independent  of  x,  must  divide  separately  the  multiplier  of  each  power  of  x.  "From  this  it 
follows  that  y-\-\  has  no  common  factor  with  B,  because,  if  it  had,  this  factor  would  be 
found  iu  y'2-^-y-^-i  and  in  y  ;  but  it  is  evident  that  >j  has  no  factor  common  with  y-\-\ 


40  ALGEBRA. 

We  can  then  multiply  A  by  y-{-l  without  affecting  the  common  divisor  sought;  and  as 
it  would  be  necessary  to  multiply  again  by  y+1,  we  multiply  at  once  by  ij/-\-l)2.  or 
y" -{-'Zy -\-\*    In  this  manner  we  arrive  at  the  remaindi  r 

R=(— y*—  y  ■'■+ //-'.  ■'■— yb— y*-\-y*- 

Before  j>assing-  to  the  second  division,  it  is  necessary  to  suppress  in  It  the  factors  com- 
mon to  the  multipliers  of  the  powers  of  a-.  But  the  two  parts  of  It  are  evidently  divisible 
by  — y* — yz-\-y-<  and  after  this  simplification  there  remains  x-\-y.  We  can  then  take 
x-\-y  for  a  divisor,  and  as  the  division  is  effected  exactly,  it  follows  that  the  common  di- 
visor sought  is  x-\-y. 

The  process  is  not  always  so  easy.  To  develop  the  general  method  to  be  pursued  in 
such  cases,  let  us  consider  the  polynomials  A  and  B,  which  contain  two  letters,  x  and  y. 
Take  first  the  greatest  monomial  common  divisor  of  the  terms  of  A;  let  a  be  this  divisor, 
and  A'  the  quotient  of  A  by  a :  we  shall  have  A=«A'.  Arrange  A  according  to  the  pow- 
ers of  a;,  taking  care  to  collect  all  the  terms  containing  a  same  power  of  this  letter,  and  let 
ns  suppose,  for  example,  that  we  have 

A/=LaB+Ma>|-N. 

All  the  factors  of  A',  independent  of  a:,  must  be  factors  of  the  quantities  L,  M,  N,  which 
multiply  the  different  powers  of  x.  These  quantities  containing  only  the  single  letter  y,  it 
will  be  easy  to  find  their  greatest  common  divisor;  let  us  name  this  divisor  a!,  and  the 
quotient  of  A'  by  a',  A";  we  shall  have  A'=a'A",  and,  consequently, 

A=aa,A". 

a  will  be  the  product  of  the  monomial  factors  of  A,  a*  the  product  of  the  polynomial  fac 
tors  which  do  not  contain  x,  and  A"  the  product  of  the  factors  which  contain  x. 

Let  us  effect  the  same  decomposition  of  the  polynomial  B,  and  let 

B=e?^B". 

Then  I  determine  the  greatest  common  divisor  of  the  monomials  a  and  ,3,  as  well  as  that 
of  the  polynomials  a'  and  ft',  which  contain  only  the  letter  y ;  and  if  I  can  also  find  that 
of  the  polynomials  A"  and  B",  which  contain  y  and  x,  I  shall  have  three  quantities,  the 
product  of  which  will  be  the  greatest  common  divisor  of  A  and  B. 

But  I  say  that  we  can  iind  thc#greatest  common  divisor  of  the  quantities  A"  and  B",  in 
subjecting  them  to  the  same  calculus  as  the  preceding  examples.  It  is  clear,  indeed,  that, 
these  quantities  having  no  longer  either  monomial  factors  or  polynomial  independent  of  x, 
it  will  be  proper  to  multiply  the  partial  dividends  of  the  first  division  by  the  polynomial 
which  is  placed  before  the  highest  power  of  a;  in  the  divisor,  and  that  we  shall  thus  arrive 
at  a  remainder  of  a  degree  less  in  x  than  the  divisor.  It  will  be  easy  to  take  from  this  re- 
mainder all  the  monomial  factors  which  it  contains,  as  well  as  the  polynomial  factors  inde- 
pendent of  a-,  and  then  proceed  to  a  second  division,  taking  for  a  divisor  this  remainder  sim- 
plified. We  operate  as  in  the  first ;  then  we  pass  to  the  third,  and  continuing  always  in 
this  manner,  we  are  sure  of  arriving  finally  at  a  remainder  zero,  or  independent  of  x. 

In  the  first  case  the  quantil  id  B''  have,  for  greatest  common  divisor,  the  divisor 

last  division. 

We  have  thus  seen  that  the  finding  of  a  common  divisor,  when  the  polynomials  contain 
two  letters,  depends  upon  finding  it  when  they  contain  one;  so  ti  here  they  con- 

tain three  depends  upon  that  where  they  contain  two,  and  so  on,  whatever  be  the  number 
of  letters. 

There  is,  therefore,  no  case  in  which  the  common  divisor  con  not  bo  found  by  the  above 
rules. 

THE  LEAST  COMMON  MULTIPLE. 
32.  We  lmvo  already  defined  a  multiple  of  a  quantity  to  be  any  quantity 
that  contains  it  exactly  ;  and  a  common  multifile  of  two  or  more  quantities  i-*  a 

ijiiantity  tl i;it  co!itaiiist«'uch  of  litem  exactly. 

*  Let  N  be  the  dh  idend,  I »  the  dh  isor,  e  the  ooefficii  at  of  the  firal  term  of  the  di\ 
Multiplying  by  the  square  of  this  coefficient,  the  dividend  The  first  term  of 

otient  will  contain  the  first  power  of  c,  Multiplying  the  whole  divisor  by  this  tern 
of  the  quotient,  every  term  of  the  product  will  contain  the  lirst  power  ^i  c,  and  the  whole 
product  maj  be  represented  by  cP.  Subtracting  this  from  the  dividend,  the  remainder  is 
c^N—  I  term  of  which  contains  <•.  and,  therefore,  its  lirst  term  is  ready  for  division 

without  multiplying  again  by  c. 


LEAST  COMMON  MULTIPLE.  41 

The  least  common  multiple,  of  two  or  more  quantities,  is,  therefore,  'Jie  leas' 
quantity  that  contains  each  of  them  exactly. 

N.  B.  The  least  common  multiple  must,  from  its  nature,  contain  all  the  dis- 
tinct factors  in  either  of  the  quantities. 

33.   To  find  the  least  common  multiple  of  two  quantities. 

(1)  Divide  the  product  of  the  two  proposed  quantities  by  their  greatest  com 
mon  measure,  and  the  quotient  is  the  least  common  multiple  of  these  quanti- 
ties ;  or  divide  one  of  the  quantities  by  their  greatest  common  measure,  and 
multiply  the  quotient  by  the  other. 

Let  a'  and  b  be  two  quantities,  d  their  greatest  common  measure,  and  m 

their  least  common  multiplo  ;  then  let 

a=hd,  and  b=kd; 

and  since  d  is  the  greatest  common  measure,  h  and  k  can  have  no  common 

factor,  and  hence  their  least  common  multiple  is  hk ;  therefore,  hkd  is  the 

least  common  multiple  of  lid  and  kd ;  hence, 

hkd3     hdxkd     axb     ab 

m=hkd= — r-= ; — = — j—=z-r  Q.  E.  D. 

d  d  d         d  ^ 

^2)  Also,  the  least  common  multiplo  is  composed  of  the  highest  powers  of  all 

the  factors  which  enter  into  the  given  quantities.* 

EXAMPLES. 

(1)  Find  the  least  common  multiple  of  2a?x  and  8a?x?. 

ab     2a2x  X  8a5x3 

Here  m=—rz= — — =8a3:r3=  least  common  multiple  : 

d  2a?x  ' 

or,  by  (2),  the  two  quantities  being  2a"x  and  23a3r5,  23a3.r3  is  the  I.  c  m. ;  be- 
cause 23  is  the  highest  power  of  2,  a?  the  highest  power  of  a,  and  r5  the 
highest  power  of  x,  in  either  of  the  given  quantities. 

(2)  Find  the  least  common  multiple  of  ±x-(x°-— y-)  and  12.r3(.r3— y3). 
Here  d=4x'2(x — y),  and,  therefore,  wo  have 

m=^= 4^=^j =l^(x+y)  (x*-tf) ; 

or,  m=12.r7-L.12a;6?/— \2xHf— \2xhf  ; 

or,  tho  two  given  quantities  being  22x2(x-{-y)(x — y)  and  22. 3.^(3: — y)(x'2-\-xy 

+y°),  the  I.  c.  m.  is  22 .  S&ix+y^x— y){z""+*y+y~)- 

(3)  Find  the  least  common  multiple  of  x2-\-2xy-\-y-  and  x5 — xy2. 
Here  d=x-\-y,  and,  therefore,  we  get 

a  ,      x"-\-2xy-\-y2  , 

d  x-\-y 

=  (x+y)(x5—xy°~) 

=x(x-\-y)  (.r2 — 2/2)=  least  common  multiple 
or,  the  two  given  quantities  being  (x-\-yY  and  x(x-^-y)  (x — y),  the  I.  c.  m.  a 
v(z+y)*(x—y). 

(4)  What  is  the   least  common  multiple  of  x4 — 5r'+9.r:! — 7.r+2,   and 
r«— 6.r2+8.r— 3? 

By  the  process  for  finding  the  greatest  common  measure,  we  find 

d=x3—3x"-+3x—l 

=  (.r— 2)  (ar*— 6a?+8z—  3) 

~x5 — 2x* — G.r3-f-20x:!  —  19.r-f-6,  the  least  common  multiple. 


The  ordinary  arlthmeuc  ;t£tn.id  depends  on  this  principle. 


42  ALGEBRA. 

(5)  Find  the  least  common  multiple  of  </-  — .V"'< -j- b9  and  a*  —  b* 

(6)  Find  the  least  common  multiple  off/-'  —  lr  and  </'-{-)•'■. 

(7)  Find  the  least  common  multiple  of  ./-'  —  </•  and  ./•  — 

(8)  Find  the  least  common  multiple  of  y* — 8y+7  a,!'1  .'/H-~.'/--8 

ANSWERS. 

(5)  (a-b)  {a*-b*).  (7)  {x+y)  (r-y  ,. 

(G)  (a— b)  (a3+  b3).  (8)  y3— 57i/+56. 

34.  Every  common  multiple  of  two  quantities,  a  anJ  b,  ?*<  a  multiple  oj  in, 
ifteir  least  common  multiple. 

For  let  ?n'  bo  a  common  multiple  of  a  and  &,  then,  because  in'  is  greater 
than  m,  if  wo  suppose  that  m'  is  not  a  multiple  of  m,  we  have,  as  in  the  an- 
nexed scheme, 

m)  m'  (ft 
to'=//to-|-Z."  ...  (1)  hm 

m! — hm=k  ...  (2)  k=  remainder. 

Now  tho  remainder  k  is  always  less  than  m  the  divisor;  hence,  since  a  and 
b  measure  m  and  m',  it  is  evident  thai  a  and  b  measure  to' — hm,  or,  by  (-2).  fc, 
therefore,  %  is  a  common  multiple  of  a  and  b,  and  it  lias  been  proved  to  be  less 
than  7n,  tho  least  common  multiple,  which  is  absurd  ;  henco  the  supposition 
that  m'  is  not  a  multiple  of  to  is  false,  or  to'  is  a  multiple  of  in. 

35.   To  find  the  least  common  multiple  of  three  or  more  quantities 

Let  a,  b,  c,  d,  Sec.,  be  tho  proposed  quantities ; 

find  to,  the  least  common  multiplo  of  a  and  b ; 

find  to',     .     « c  and  m  ; 

find  to", d  and  m' ;  cVc. 

The  last,  say  m",  is  evidently  a  multiple  of  a,  b,  c,  d,  &c. 
Then,  since  every  multiplo  of  a  and  b  is  a  multiple  of  m,  their  least  common 
multiple  (34),  the  quantity  sought,  x,  is  a  multiple  of  to;  but  x  also  is  by  hy- 
pothesis a  multiple  of  c  ;  therefore,  x  is  a  multiplo  of  both  c  and  m,  and,  there- 
fore, it  is  (34)  a  multiple  of  to'  ;  but  x  is  a  multiplo  of  d  and  to',  and,  therefore, 
of  to"  ;  hence  x  can  not  bo  less  than  m",  and,  therefore,  to"  is  tho  least  com- 
mon multiple. 

EXAMPLES. 

(1)  Find  tho  least  common  multiple  of  2a9,  40*6*,  and  Gab3. 
Hero,  taking  2a*  and  Aa3lr,  wo  find  d*=2a9,  and,  therefore, 

ab     2a«X4fls6a 

d 

Again,  taking  m,  or  Aa3b",  and  G/z/j3,  we  find  /<7=2a/V;  hence 

cm     Gai3x4" 

7n,'=—r= — x— r; =12a3b3=  answer  required. 

a  2ao" 

Or,  the  three  given  quantities  beini  •  and  2.3ao',  the  2. cm.,  by 

<J3.(2),  is2*.3</ "/<  . 

(2)  Find  the  least  common  multiple  of  a—  nd  </'■—,  . 

Taking  a — C  and  r/: — .;-',  we  have  d  =  a — X;   and  henco 

ab     a — x 

in=-t= x  ((/•—./-')  =  •    —     ■ 


— =(a-{-x)(a5— ar5)=  answer  sought. 


FRACTIONS.  43 

Again,  taking  a2— a:2  and  a3— x3,  we  find  d=a—x;  hence 
cm      (a3— 2J)(a2— s2) 
a!  a — a; 

Or,  the  three  given  quantities  being  (a  — x),  (a—x)(a-\-x),  and  {a—x^cp+ax 
+x"-),  the  least  common  multiple  is  {a—x){a-\-x){a?-\-ax-\-x"). 

(3)  Find  the  least  common  multiple  of  15a262,  12ab3,  and  6a36. 

(4)  Find  the  least  common  multiple  of  6a2x"(a— x),  8x3(a'2— x2),  ard  12 
{a—xy. 

(5)  Find  the  least  common  multiple  of  x3— x-y— xy*+y3,  o^—x-y-^xif—y3, 
and  x4 — y4. 

(6)  Find  the  least  common  multiple  of  {a-\-bf,  (a2  —  P),  (a  — by,  and  a3 
+  3a'2b  +  3a¥-\-b3. 

(7)  Find  the  least  common  multiple  of  45,  50,  and  75,  or  32 . 5,  2 .  52,  and 
3.52. 


ANSWERS. 

(3)  60a3b3.  (6)  (a+b){a*—b*y 

(4)  24a2x3(a— z)(a2— a*). 

(5)  x5 — a:i/4 — x4^/+2/5. 


(7)  32.2.52=450. 


OF   ALGEBRAIC  FRACTIONS. 

36.  Algebraic  fractions  differ  in  no  respect  from  arithmetical  fractions,  and 
all  the  rules  for  the  latter  apply  equally  to  the  former.  We  shall,  therefore, 
merely  repeat  the  rules,  adding  a  few  examples  of  the  application  of  each.  It 
may  be  proper  to  remind  the  reader  that  all  operations  with  regai-d  to  frac- 
tions aro  founded  upon  the  three  following  principles  : 

1.  In  order  to  multiply  a  fraction  by  any  number,  we  must  multiply  (lit- 
numerator,  or  divide  the  denominator  of  the  fraction  by  that  number. 

2.  In  order  to  divide  a  fraction  by  any  number,  we  must  divide  the  numera  ■ 
tor,  or  multiply  the  denominator  of  the  fraction  by  that  number. 

3.  The  value  of  a  fraction  is  not  changed,  if  we  multiply  or  divide  both  the 
numerator  <lnd  denominator  by  the  same  number. — See  (17,  Note). 

REDUCTION  OF  FRACTIONS. 
I.   To  reduce  a  fraction  to  its  loioest  terms. 

37.  Rule. — Divide  both  numerator  and  denominator  by  their  greatest  com- 
mon measure,  and  the  result  will  be  the  fraction  in  its  loioest  terms. 

When  the  numerator  and  denominator  are,  one  or  both  of  them,  monomials, 
their  greatest  common  factor  is  immediately  detected  by  inspection ;  thus 
a"bc       a"b  X  c        c 


=—r  in  its  lowest  terms. 


So,  also, 


5a"b~~aibx5b~5b 


ax"2  x  X  ax         ax 

-= — ; —  in  its  lowest  terms. 


ax-\-x2     x(a-{-x)     a-\-x 
If,  however,  both  numerator  and  denominator  are  polynomials,  we  must 
have  recourse  to  the  method  of  finding  the  greatest  common  measure  of  two 
algebraic  quantities,  developed  in  a  former  article.     Thus,  let  it  be  required  to 
reduce  the  following  fraction  to  its  lowest  terms  : 


44  ALGEBHA. 

6a3— 6a2  y -4- 2ay- — If 

12as —  15«i/4-%" 
The  greatest  common  measure  of  the  two  terms  of  this  fraction  was  found  at 
page  37  to  be  a — y;  therefore,  dividing  both  numerator  and  dencminator  by 
this  quantity,  wo  obtain  as  our  result  the  fraction  in  its  lowesl  tarns;  or, 

Ga-+.' 

12a— 3?/' 

t    ri  *i-      .i      e     *•  4a<— 4a5i24-4«/.— M  . 

In  like  manner,  taking  tho  fraction JL ,  the  greatest 

i_j_4a3j_  Qasjs_  ZaP+ib* 

common  measure  of  tho  two  terms  is  found  to  be  2a?-\-2ab — ft9;  and,  drridi  ^ 
both  numerator  and  denominator  by  this  quantity,  the  reduced  fraction  is 

2a3— 2ab+   b* 

3a2—  ab—2b2' 

EXAMPLES   FOR  PRACTICE. 

2.r3— lG.r— 6 

(1)  Reduce  ■ _    to  its  lowest  terms. 

48r5+36.r2— 15 

(2)  Reduce  — -= — ,.,  „  ,  ,  „ -  to  its  lowest  terms. 

v  '  24ar! — 2l2--f-l&r — 0 

20.r'4-.r-  — 1 

(3)  Reduce  nr  ,  ,  .  ., to  its  lowest  terms. 

'  25r'+5ri — x — 1 

/      -r.    ,        3m-/i  —  m:  i)  —  6mn*-\-2mnp 

(4)  Reduco  -— t-t-z — — to  its  lowest  terms. 

'  12»m —  15ns — 4mp-\-on2> 

m9 — 2mn 

Ans.  — : — . 

4m — on 

(5\   Redu       4aflcx—4aZdx+2ia~bcx—24a*bdx-\-36ab2cx—36al-tljc 

'  labcxS—labdx*+lcufix*—1acdx*—'Zlb-dx*  +  Klb*afl-{-%\b&x*—  %  1  betW 

.     ,  4a(a  +  36) 

to  its  lowest  terms.  Ans.  — rr-i -■ 

7x*(b-\-c) 

38.  It  frequently  happens,  however,  that  when  tho  polynomials  which  form 
the  numerator  and  denominator  of  a  fraction  which  can  be  decomposed  are  not 
veiy  complicated,  wo  are  enabled  by  a  little  practice  to  detect  the  common 
factor  and  effect  the  reduction  without  performing  the  operation  of  finding  the 
greatest,  common  measure,  which  is  generally  a  tedious  process.  The  resulni 
to  which  wo  called  the  attention  of  the  reader  at  the  end  of  algebraic  dh 
(see  pago  30)  will  bo  found  particularly  useful  in  simplifications  of  this  nature. 

Thus,  for  example  : 

"■''.'/(•''+.'/)  ( ■''+.'/) 

3x»+6xy+^-  3(z+y)«  =2{x+y){x+y)-x+y 

„■_/,-•         (a— &)(«+&)     a+6 


(G) 
(7) 
(8) 

(») 

(10) 


d: — 2ab-{-b1  (<i  —  /»)J         n — V 

oa*+l0a"-h+5ah*    5a(a  +2a6+o»)    5a(a-fr b)9    o{a+b) 

8a3+8</ ■■/>  \-bj~  (<i  +  l>)~  ~^i      ' 

an—  ._L.T-)(a— x)     at+ax+x* 

d»— 2ax+:rB=  (a—  a—  x 

ac+bd+ad+bc    _  {a+b)c+(a+b)d  _  (,■  +  ,/)(,/  +  h)  _  c+d 
af+  2bx+2cu  +  bf~{a  + 1  \J'+  2 1  (a  +  b)~  (/+  2  c)(a  -f  b) -/+  2  c 


TRACTIONS.  41 

6ac+Wbc+9ad+15bd  3a(2c+3d)+5b(2c-\-3d)_(3a  +  y,)(2r+3d) 
(U)  6c->+9cd—2c—3d  ~  3c(2c+3d)—(2c+3d)  ~(3c—V){^c~^3a) 
3a+5b 


3c— 1 

axm— bxm+1       xm(a — bx)  xm(a — bx)  xm~l 

(12) 

(13) 

(14) 


cPbx—bPx3      bx(a*—b*x*)     bx(a-{-bx){a — bx)     b(a-\-bx) 
a4—b*  _a2— &2 
a?-\-ab'z         a 

2xy + 3f + 2x" + 3xy  y + x 

8cx+\2cy—lQdx—lhdij~lc—bd' 


II.   To  reduce  a  mixed  quantity  to  an  improper  fraction. 

39.  Rule. — Multiply  the  integral  part  by  the  denominator  of  the  fraction, 
and  to  the  product  add  the  numerator  with  its  proper  sign ;  then  the  result 
placed  over  the  denominator  ivill  give  the  improper  fraction  required.     Thug 

a  a4-b 

a2— x'i_a?-\-X'-\-a"— x- _     2a2 
™  1"^a24-2-3==        aF+x*        =a?+x2' 

,     abc—c°d—2cd*     abc+c"d+2abd+2cd24-abc—c°~d—2cdi 

(3>  ab+cd+         e+2d         =~^^ 7+21T 

2abc+2abd 
—  c+2d 

2ab(c+d) 
=  c+2d 

&2_|_C2_a<2     goc-ffo+c8— fl8     (ft-f-c)8— a8 
(4)  ^        2bc        =  26c"  =        2bc        ' 

40.  It  is  to  be  remarked  that  when  a  fraction  has  the  sign  — ,  it  signifies 
that  the  whole  fraction  is  to  be  subtracted;  the  negative  sign,  therefore,  as 
will  bo  shown  hereafter,  applies  to  the  numerator  alone  ;  when  the  numerator 
is  a  polynomial,  the  negative  sign  extends  to  all  its  terms;  the  bar  which  sepa 
rates  the  numerator  from  the  denominator  is  to  be  regarded  as  a  vinculum,  and 
if  it  have  the  negative  sign  before  it,  when  removed,  all  the  signs  of  the  numer- 
ator must  be  changed. 


(5)  1- 

(6)  c 

(7)  1 
8)1 


b      a — b 


a         a 
ef     cd — ef 
~d  =     d     - 

as— 2a6  +  fr»     ar+b2— (a2— 2ab+b9)       Sab 
a9+6»  a-+62  =a--f-K 

5g4.cs— q»     25c  —  (fr2-f  c2— a~) 
2bc       ~  2bc 

a?— (58— 26e-f-ca) 

"~  26c 

_a2— (b—  c)2 
"~        2bc 


46  ALGEBRA. 

x3  —  3x*y +  ?>.r  if  —  if 


(9)  x*+2xy+if- 


x+y 
■t3  -f  3.r- y  +  r.rif  +  if—(x3  — 3x-y  -f  3ry"—f) 

Gx-y  +  2y3 


x+y 
J>y(3x-+if) 
x+y 

2mw2 — 2pqn     m-n — mpq+mn2 — npq —  (2mn'i —  2pqn, 

10)  mn—pq— rJLJ-  = OI!= r1 — S LL2 

'  s  1  m-\-n  m-\-n 

m-n — mpq — mn-+pqn 

m+n 

mn(m — n) — pq(m — n) 

m  +  n 

(mn — pq){rn — n) 

m-\-n* 

ill.   To  reverse  this  process,  or  to  reduce  an  improper  fraction  to  a  m 
quantity. 

Rule — Divide  the  numerator  by  the  denominator  ;  the  quotient  obtained  m 
far  as  practicable,  will  be  the  entire  part,  and  the  remainder,  set  over  /' 
nominator,  will  be  the  fractional  part.     Then  the  two,  joined  tog' 
proper  sign,  will  form  the  mixed  quantity  required.     Thus, 

ay+b  b 

11)  -^-=0+-. 

a24-x2  2x2 

(12)  •— - — =a+x+ . 

v     '    a — x  '       '  a — x 

20X3— 10.r+4  4 

„  .,  p°+2pq  +  q°—r—s  r+s 

(14) JV, =F+q-¥+S 

mHm*— n*)  +  3m3— 3mn-  3 

(15)  — * ttt — JT =  m"+n-+- . 

v     '  m-(iit-  —  ;j2)  '  m 

IV.  To  reduce  fractions  to  others  equivalent,  and  having  a  common  denomi- 
nator. 

41.  Rule. — Multiply  each  of  the  numerators,  separately,  into  all  the  denomi- 
nators, except  its  ou-n,for  the  new  numerators,  and  all  tiie  dt nominators  to 
gether  for  a  common  denominator.] 

a         c 
Thus,  reduce  r  and  —.  to  equivalent  fractious  having  a  common  denominator 

a  X  d  is  the  new  numerator  of  the  Bret, 

c  x  b  is  the  new  numerator  of  the  second, 
bxd  is  the  common  denominator; 

ad  be 

Therefore,  the  fractions  required  are  ,   ,  and  —.. 

*  Tin;  rationale  of  the  above  examples  is  given  in  the  note  on  the  next  page. 

t  The  numerator  and  denominator  of  each  fraction  will  thus  l>e  multiplied  by  the  same 
number,  viz.,  the  product  of  the  other  denominatora,  nml,  oonseqoently,  "ta  value  will  be  on 
■hanged* 


FRACTIONS.  47 

^^accgkm 

.Reduce  -r,  "?>  7>  T>  T>  —  >  to  a  common  denominator. 
b   a  J    hi    n 

adfhln    cbfhln    ebdhln  gbdfln  Jcbdfhn  mbdfhl  .      _  . 

bdfhln   bdfhln   bdfhln    bdfhln    bdjldn    bdfhln 

\-i-x   14-x2   14-ar5  ' 

Reduce  — !— ,  — JL — ,  __ L__,  to  a  common  denominator. 
I — x  1 — x2  1 — x3 

(!+*)(! -*»)(! -r>)    (1 +*"')(!  -x)(l—r*)    (l+r»)(l  -x)(l-x2) 

(l_a:)(l_x2)(i_r5)'   (1— x)(l  —  x2)(l  —  xa)'   (l_x)(l— x2)(l— x3)'  ' 

fractions  required.  , 

ADDITION  OF  FRACTIONS. 
42.  Rule. — Reduce  the  fractions  to  a  common  dencr.iinalor,  add  thenumera 
tors  together,  and  subscribe  the  common  denominator.     Thus, 
a      c      ad      be     ad-\-bc 
^  1+d—bd+bd==     bd     ' 

a     m     p      x     anqy     mbqy     pbny     xbnq 
'   b'*~n'q*y      bnqy*bnqy'bnqy*bnqy 
anqy  -\-  mbqy -(- pbny -f- xbnq 
>  bnqy 

ace      adfx5     cbfx*      ebdx* 
^  bl:^dx^fxs=bdfx^^r  bdfx^  bdftf 
adfi^-\-  befx*  +  bdex3 

=         bdfi*        ' 

1+3*      1-S»  (1  +  X2)2  (!-■ *T 

W  l _x3+ 1  +  x-~~ (1  — x2)(l+x2)  +  (1  — x2)(l  +x2) 

(l+X2)2+(l-X2)2 
2(1 4-  X*) 


1  _1 1— X  1  +  X 

^  l+x"*~l—x— (l+x)(l— x)  +  (l-fx)(l— x) 
_1— x+l+x 
-(l+x)(l-x) 
2 


1— x2 

SUBTRACTION  OF  FRACTIONS. 
43.  Rule. — Reduce  the  fractions  to  a  common  denominator,  tubtia.**  tfu 
numerator  or  the  sum  of  the  numerators  of  the  fractions  to  be  subtracted,  from 
the  numerator  or  the  sum  of  the  numerators  of  the  others,  and  subscribe  the  com- 
mon denominator.* 

a      c     ad      be      ad — be 
(1'  b~d=bd~bd—     bd     ' 

am      ip     x\      anqy     mbqy     2ybnV     xbnq 
'   6  •"  n      \q     y)      bnqy*   bnqy      bnqy     bnqy 
anqy  -\-mbqy — pbny — xbnq 
bnqy 

*  The  rales  for  addition  and  subtraction  of  fractions  follow  from  the  general  principle  thai 
quantities  to  be  added  or  subtracted  must  be  of  the  same  denomination. 


48  ALGEBRA 

a       c        e       g      ad/Jufl      bcfJufl      h  d)   '      bdfgz? 
^  bx+dx2 ~Jxz~kxi==  bdfhx^ bjfhx10 ~~ bdfhxl0~ bdf  t> 

adfhs?  -\-hrfh ./•-  — bedkx    — bdfg 
—  bdfhx* 


\*l 

a  —  b 

«  +  6" 

~     (a  +  b){a—b) 
Aab 

-a?—b~' 

(5) 

1+x2 

1  — x2 

(1+x2)2 

(1- 

-X2)2 

1  — x- 

1+x2 

-(l_a*)(l+a*) 

(1- 

—  X 

»)(!+: 

*) 

(l+x2)2-(l- 

x2)2 

~     (l-x2)(l  +  x 

') 

4x2 

-1— X*' 

(6) 

1 

am-n 

1       a' 
a'" 

'  — 1 
am 

a2         frVl      a2Zr+2— Z^o7 

44.  When  the  denominators  of  the  fractions  which  it  is  required  to  reduce 
have  a  common  multiple  less  than  their  continued  product,  the  result  will  fre- 
quently be  much  simplified  by  finding  this  least  common  multiple,  and  then 
reducing  the  fractions  to  their  least  common  denominator  by  multiplying  the 
numerator  and  denominator  of  each  fraction  by  the  quotient  of  the  least  com- 
mon multiple,  divided  by  the  denominator  of  that  fraction. 
Thus,  if  we  are  required  to  reduce  the  following  fractions  : 

a — 3x     3a — 5x     3a — 5x 
4~  "*"      5        '       20     ' 
The  least  common  multiple  of  4  and  5  is  20,  the  denominator  of  the  thirJ 
fraction  ;  therefore  the  fractions,  when  reduced  to  their  least  common  denoini 
uator,  are 

5a  — 15x     12a— 20x     3a— 5x_5a— 15x-j-12a— 20x+3a— 5x 
20      "*"       20        ~^~~ 20      =  20 

20a — In 


20 

=  a  —  2 
So,  also,  in 

27— 9x     5x+2      CI      2x-f5     29+4*     5— 37x 
X^"     4  G  12"*"     3      "•"     12  12     ' 

tho  least  common  multiple  of  3,  4,  G  is  12,  which  will  be  the  least  common  de>- 
nominator,  ;mcl  the  above  fractions  become 

12x     81— 27x     10x+4      (11      Rr+20     29+4x     5— 37x 
12 T      12  L2  12T      12      T     12  12 

Or, 
12x+81— 27x— lOx— 4— 61  +  8x-f  21 I -f  -" 1 4-  lx— 5-f  37x_24x-f  GO 

12  ~~  ~~ i~2 

=2r+5. 


FRACTIONS.  4!» 

MULTIPLICATION  OF  FRACTIONS. 
45.  Rule. — Multiply  all  the  numerators  together  for  a  new  numerator,  and 
til  the  denominators  togeOierfor  a  new  denominator.     Thus,* 
a     c      ac 

W  bxd^bd- 

a     m      v     x     ampx 
'2)  aX-X^X-= 


a 


q     y      bnqy ' 


a+b  ■   e-£      k+l      p-q     (a+b)(e-f)(k+l){p-q) 
3)   c-\-dXg-hXm-nX  r+s-{c+d)(g-h)(m-n){r+sy 

abode        abcde         a 
U)  bl:Xcx^Xa^XexiXJr^=bcdefx15=JxTi' 

DIVISION  OF  FRACTIONS. 

AG.  Rule. — Invert  the  divisor  and  proceed  as  in  Multiplication.^ 
a      c     a     d     ad  c     ad     acd'.   a 

W  b+d=bx-c=-b7-  Proof'  dxTc=bTd=b- 

{">  c+d-g-h-c+dX  e-f~(c+d){e-fy 

l+g-3  _  1—  a8     1+a*     1  +  a?     (1+x2)2 
*3)  IZZ^l+x2- l_a:2><l— x2_  (1— z2)2" 

x* — b*  x'2^-bx  .r4 — b*  x — b 

^  xq—2bx-\rb'i~r  x—b  =x2— 2bx-\-biX  x^+bx' 

(xi—b*)(x—b) 


'(a?— 2bx+b*)(x*+bx) 

(3*— &3)(a«+68)(j— ft) 
=     (X—by.x.{x-\-b) 

(x+b)(xg-  b)(x"-\-b%x— b) 
=       x(x—b)(x—b)(x+b) 

x2+ft2 


47.  Miscellaneous  Examples  in  the  operations  performed  in  Algebraic  Frae 
>ions.  ^ 

3a     hf      x      42aey-\-35bfy—8bex 
^  Tb^r8e~7y~  56bcy 

2a      bdf     deg      16abc-\-15cdf—4deg 


(2) 


3bc~8b-c     6b"c-~  24ft2c2 


/«         f     g3   ,  fm      6c/g(e-/)-3g«+2/W 
W  e-J-2ef  +  3eg-  6efg 

a        c  dg        a — cx+G?xr+8 

Tn        Xn — 1~^Xn — r — a  Xn 


*  To  multiply  a  quantity  by  the  fraction  3,  for  instance,  is  to  take  it  as  many  times  as  is 
expressed  by  this  multiplier,  that  is,  two  thirds  of  a  time,  or  to  take  two  thirds  of  it, 
which  is  done  by  dividing  it  by  3,  and  multiplying  by  2.  If  the  multiplicand  be  a  fraction, 
this  is  done,  as  has  been  before  shown  (17,  Note),  by  multiplying  its  numerator  by  2,  and 
its  denominator  by  3,  which  accords  with  the  rule  above  given. 

t  This  rule  depends  upon  the  principle  that  the  divisor,  multiplied  by  the  quotient,  must 
produce  the  dividend. 

D 


« 


50  ALGEBRA. 

b"c— 5a62c+a3     2a63— 6c2+3a6cs— a* 


62 — 6c  6- — be 


(5)  c+2a6  — 3ac- 

(6)  -g — I — o~=a 
a+6     a-b 

(')     -Q- Q~  =b- 

13a— 5b     7a— 2b     3a     89a— 556 


(9) 


4  6  5  60 

3a— 46     2a  — 6— c     15a— 4c     85a— 206 
7      ~        3        "*"       12      =       84       ' 


a     a— 36     a2— 62— a6     aca7— 462+a3 
(1°)  6+~ c~d~+        bed        =  bc~d         * 

a3  a6  _6 a3+a62+63 

(U>  (a+6)3_(a+6)2+a+6_    (a+6)3     ' 

(12)  3  ,  3  1        _     1-x  l  +  x+X» 

v     '  4(1—  x)2T8(l  —  x)^8(l+x)      4(l+x2)- 1-  x  -x'+x*' 

V1^  a2— 62Xa+6— a2+2a6+62* 

.        x2— 9x+20     a:2— 13x+42     a:2— llx+28 

*     '       x2— 6x      X       a:2— 5.r      =  5  * 

x2+3x+2      x2+5x+4      x+2  ^ 
V     '  x2+2x+lXx2+7x+12— x+3* 

a     c 

b  +  d     (ad+be)fh 

e,£~(eh+fs)bd' 

a  b 

a+b  +  a~=b 


a — 6     a+6 

a+x     a — x 

a — x  '  a+x     a2+x2 

(ic\  ! ' 

"     '  a+x     a — x       2ax 
a — x     a+x 

*-l 

<19>  — £r 
i 


<20)^ 


71+1 

a3_a-x+ax2— x3  a4— x« 


a6 — aAx-\-a?x- — a^+ax* — x*     a6 — x6 

a'+x« 


"o+aV+i*' 


•  These  examples  admit  of  tho  application  of  the  formulas  at  the  top  of  p.i 


EXTRACTION  OF  HOOTS.  51 

ON  THE  FORMATION  OF  POWERS,  AND  THE  EXTRACTION 
OF  ROOTS  OF  ALGEBRAIC  QUANTITIES. 

48.  We  begin  by  considering  the  case  of  monomials,  and,  in  order  to  sim- 
plify the  subject  as  much  as  possible,  we  shall  first  treat  of  the  formation  of  the 
square  and  the  extraction  of  the  square  root  only,  and  then  proceed  to  gener- 
alize our  reasonings  in  such  a  manner  as  to  embrace  powers  and  roots  of  any 
degree  whatsoever. 

Definition. — The  square  root  of  any  expression  is  that  quantity  which, 
when  multiplied  by  itself,  will  produce  the  proposed  expression.  Thus,  the 
square  root  of  a~  is  a,  because  a,  when  multiplied  by  itself,  produces  a" ;  the 
square  root  of  (a-\-b)-  is  a-\-b,  because  a-\-b,  when  multiplied  by  itself,  pro- 
duces (a-\-b)2  ;  in  like  manner,  8  is  the  square  root  of  64,  12  of  144,  and  so  on. 
The  process  of  finding  the  square  root  of  any  quantity  is  called  the  extraction 
of  the  square  root. 

The  extraction  of  the  square  root  is  indicated  by  prefixing  the  symbol  \f  to 
the  quantity  whose  root  is  required.  Thus,  V  a1  signifies  that  the  square  root 
of  a*  is  to  be  extracted  ;  -\/a--\-2ab-\-b2,  or  V (a2 -\-2ab +&2)»  signifies  that  the 
square  root  of  a"-\-2ab-\-bi  is  to  be  extracted,  &c. 

In  order  to  discover  the  method  which  we  must  pursue  to  extract  the  square 
root  of  a  monomial,  let  us  consider  in  what  manner  we  form  its  square.     Ac 
cording  to  the  rule  for  the  multiplication  of  monomials, 

(5a"b3cY=5a'bsc  X  5a263c=25aW. 

So, 

(9a62c3J-,)2=9a52c3^  X  9ab"-c3d4=81a-b*c6d8. 

And, 

(Axmynzh  -  -  -)-=Axmynzb  -  -  -  X  Axmyazh  -  -  -= A2a:2ayz2h  -  -  -  ; 
i.  e  ,  we  add  the  exponent  of  each  letter  of  the  given  monomial  to  itself. 

49.  Hence  it  appears  that,  in  order  to  square  a  monomial,  we  must  square 
its  coefficie/tt,  and  midtiply  the  exponents  of  each  of  the  different  letters  by  2. 
Therefore,  in  order  to  derive  the  square  root  of  a  monomial  from  its  square, 
we  must, 

I.  Extract  the  square  root  of  its  coefficient  according  to  the  rules  of  Arith- 
metic. 

II.  Divide  each  of  the  exponents  by  2. 
Thus,  we  shall  have 

V64a664=8a362. 
This  is  manifestly  the  true  result,  for 

(8a362)2=8a362  X  8a362=64a6fr4. 
Similarly, 

Here,  also. 
Again, 


•v/625a268c6=25a64f3. 
{25abic3)"z=.25ab4c3  X  25ab4c5=625at't*ce 

V  25a6p-18c4d-32 =5a3p~9c2  d~lG —-^ 


Also, 


■y/8la2mximy6nz*r~~i=9amx~my3nzv-1. 


52  ALGEBRA 

\lso, 

m     rk+p — q     1 

\fl6cmdn+P-ig=4c*d    *    gi. 
if  the  given  quantity  bo  a  fraction,  extract  the  square  root  of  its  numerator 
and  denominator  separately.     This  rule  follows  from  that  for  multiplication 
of  fractions.     Thus, 

Il9a*be_7a-b3 

VlGc2^4-  4cd*' 

Also,  

l36a-mbSa     6amb*n 
V  64a2Pc4  =  8aPcs  " 
\lso, 

I     a^c*  a?b5c 


V  (a + z)  -/i1  Y     (a + x)  /i53/a' 
Also, 

•>-2ii3 


V  \  m4    X  a^4/8/  "afflW/** 


50.  It  appears,  from  the  preceding  rule,  that  a  monomial  can  not  be  the  squaie 
of  another  monomial  unless  its  coefficient  be  a  square  number,  and  the  exponents 
of  the  different  letters  all  even  numbers.  Thus,  98ab*  is  not  a  perfect  square, 
for  98  is  not  a  square  number,  and  the  exponent  of  a  is  not  an  even  number. 
In  this  case  we  introduce  the  quantity  into  our  calculations  affected  with  the 
sign  VTand  it  is  written  under  the  form  */98ab*.  Expressions  of  this  nature 
are  called  Surds,  or  Radicals  of  the  Second  Degree.* 

51.  Such  expressions  can  frequently  be  simplified  by  the  application  of  the 
following  principle  :   The  square  root  of  the  product  of  two  or  more  factors  i$ 
equal  to  the  product  of  the  square  roots  of  these  factors.     Or,  in  algebraic  Ian 
guage,  _         _         _        _ 

y/abed =  y/aX  y/°X  V?X  Vd 

In  order  to  demonstrate  this  principle,  let  us  remark  that,  according  to  our 
definition  of  the  square  root  of  any  expression,  wo  have  „ 

(  y/abed )-=abcd 

Again^ 

(V^x  Vox  Vex  Vd---y=(V<i):x(Vt>Y-x(Vcyx(Vdy~-i 

ssdbcd . 

Hence,  since  the  squares  of  the  quantities   \Jabed ,  and  y/a-  V° 

y/c  y/d —  are  equal,  the  quantities  themselves  must  be  equal. 

This  being  established,  the  expression  given  above,  v98ao*,  may  be  put 
under  the  form  ^49i«X2a==  y/^'TX  V~^,  but  y/Ml>*  >9  by  (Art.  49)=75» ; 
hence 

y/98b*a=  yflbT*  x  i/2a=7&»  \J%0. 

Similarly,  

y/TbaW<*ds*  y/9a'!l>-ciXbt>d=  \Z9n"6-c3  X  y/^d 

=  3abcy/5bd- 

*  From  tho  Latin  turd/Ut.  They  arc  sometimes  called  inconimcnsuraWe,  having  no  com- 
mon measure  with  unity.  They  aro  also  called  irrational,  because  their  ratio  with  unity 
ota  in  t  be  expressed  in  numbers.  Fractions  have  Kith  a  common  measure  and  ratio  with 
unity,  Tim i  the  frax  don  j  has  }  for  a  rorumon  measure  w  ith  unity,  and  Its  ratio  with  <mi 
tv  is  I.  t  This  I  m  (10,  111.,  note). 


EXTRACTION  OF  ROOTS.  53 

^o,  also, 


Also, 
Also, 
Also, 
Also, 
Also, 


l/24a"b=! 


-v/54a363c=3a&  y/tiabc. 


2  A/8a2,n+16=4am  y/2ab. 


3  y/75p9q4=15Fif  Vty. 

g    /48.r2i'+3?y3      SG-tp+'t/2    /3x?/      18.i't'+1_y2    /n/ 
V"  12a26    =      2a      V  ~3T =       a      V  "6"' 


In  general,  therefore,  in  order  to  simplify  a  monomial  radical  of  the  second 
degree,  separate,  those  factors  which  are  perfect  squares,  extract  their  not  (Art. 
49),  place  (lie  product  of  all  these  roots  before  the  radical  sign,  and  place  all 
those  factors  which  are  not  perfect  squares  under  the  radical  sign. 

In  the  expressions,  lb'ly/2a,  3abc-\/5bd,  12ab2c5  V 6bc,  &c,  the  quantities 
7b2,  3abc,  12ai2c5,  are  called  the  coefficients  of  the  radical. 

52.  We  have  not  hitherto  considered  the  sign  with  which  the  radical  may 
be  affected.  But  since,  as  will  be  seen  hereafter,  in  the  solution  of  problems 
we  are  led  to  consider  monomials  affected  with  the  sign  — ,  as  well  as  the 
sign  -J-,  it  is  necessary  that  we  should  know  how  to  treat  such  quantities. 
Now  the  square  of  a  monomial  being  the  product  of  the  monomial  by  itself,  it 
necessarily  follows  that,  whatever  may  be  the  sign  of  a  monomial,  its  square 
must  bs  affected  with  the  sign  +.  Thus,  the  square  of  -\-ba?b3,  or  of  — 5a2b3, 
is  +  25a466. 

Hence  we  conclude  that,  if  a  monomial  be  positive,  its  square  root  may  be 
either  positive  or  negative.  Thus,  ■\/9a4=-\-3a!i,  or  — 3a2,  for  either  of  these 
quantities,  when  multiplied  by  itself,  produces  9a4 ;  we  therefore  always  affect 
the  square  root  of  a  quantity  with  the  double  sign  rt ,  which  is  called  plus  or 
minus.     Thus,  •/9a*=  ± 3a2,  71141^6=  ±12aZ>2c3.* 

53.  If  the  monomial  be  affected  with  a  negative  sign,  the  extraction  of  ita 
square  root  is  impossible,  since  we  have  just  seen  that  the  square  of  every 
quantity,  whether  positive  or  negative,  is  essentially  positive.     Thus,   sf — 9, 

*  The  double  sign  may  be  omitted,  being  always  understood  before  y/ .  An  important 
proposition,  not  usually  noticed,  should  be  demonstrated  here ;  it  is,  that  the  quantity  A  has 
no  otber  square  root  than  the  two,  -\-\/  h.  and  — -j/A.  To  prove  this,  let  us  observe  that 
the  different  square  roots  of  A  are  the  values  of  x  in  the  equation  x2:=A,  or  what  is  the 
same, 

x'i—K—Q. 

Instead  of  x2 — A,  we  may  write  x" — (1/A)2 ;  then,  decomposing  this  difference  into  two 
factors,  we  have 

x2— A=  (x—  V A)  (x-f-i/A). 

Under  this  fonn  we  perceive  that  every  value  of  x  which  is  not  either  +V'A  or  — \ZA, 
will  fail  to  render  either  of  these  two  factors  zero ;  then  it  will  not  render  the  product  x- — A 
•zero ;  therefore  the  quantity  A  has  no  other  square  root  than  ^y/A. 

The  square  root  of  a  quantity  lias,  therefore,  two  values,  which  are  equal  with  contrary 
*ign$  and  it  has  no  otlier  values. 


54  ALGEBRA. 


•\/ — Adz,  -/ — 5,  are  algebraic  symbols  which  represent  operations  which  it  is 
impossible  to  execute.  Quantities  of  this  nature  are  called  imaginary  or  im- 
possible quantities,  and  are  symbols  of  absurdity  which  we  frequently  meet 
with  in  resolving  quadratic  equations. 

By  an  extension  of  our  principles,  however,  we  perform  the  same  opera 
tions  upon  quantities  of  this  nature  as  upon  ordinary  surds.  Thus,  by  (Art 
51), 

V~^9      =  y/9X—  1  ==  -/?■  V~l  =3  V"l. 

V— 4£_=  V^a- X  —  1  =  yfia?  V~—i  =2a  -/  —  1  _ 

V  —  8a26=  -/^X4a-X^X— 1=  i/ia=  X  V~2b  X  V—  1  =2a  \f  26  V  —1 

54.  Let  us  now  proceed  to  consider  the  formation  of  powers  and  extraction 
of  roots  of  any  degree  in  monomial  algebraic  quantities. 

Definition. — The  cube  root  of  any  expression  is  that  quantity  which,  mul 
tiplied  twice  by  itself,  or  taken  three  times  as  a  factor,  will  produce  the  pro 
posed  expression.     The  fourth,  or  biquadrate,  root  of  any  expression  is  that 
quantity  which,  multiplied  three  times  by  itself,  or  taken  four  times  as  a  fac 
tor,  will  produce  the  proposed  expression ;  and  in  general,  the  n&  root  of  any 
expression  is  that  quantity  which,  multiplied  (n  —  1)  times  by  itself,  or  taken 
n  times  as  a  factor,  will  produce  the  proposed  expression.     Thus,  the  cube 
root  of  a?b3  is  ab,  because  ab,  multiplied  by  itself  twice,  or  taken  three  times 
as  a  factor,  produces  a3Z>3;  for  the  same  reason,  {a-\-b)  is  the  6th  root  of 
(a-\-b)6,  2  is  the  seventh  root  of  128,  and  so  on. 

55.  Let  it  be  required  to  form  the  fifth  power  of  2a3b2. 

i2a3b-y=2a3b"-  X  2a*b2  X  2a3bi  X  2a3i':  X  2a362 
=32a156l°. 
Where  we  perceive,  1°.  That  the  coefficient  has  been  raised  to  the  fifth 
power ;  2°.  That  the  exponent  of  each  of  the  letters  has  been  multiplied  by  5 
Tn  like  manner, 

(8a2isc)3=8a263c  X  8a263c  X  8a2i3c 

=  83a2+2  +2£3+3+3cl  +1+1 

=512u6i9c3. 
So,  also, 

(2ab°chli)n=2ab"-cWx2ab2c3d-iX to  n  factors 

=2nanb-nc3"din. 
Hence  we  deduce  the  following  general 

RULE  TO   RAISE  A  MONOMIAL  TO  ANT  POWER. 

Raise  the  numerical  coefficient  to  the  given  power,  and  multiply  the  exponents 
of  each  of  the  letters  by  the  index  of  the  power  required.* 
And  hence,  reciprocally,  wo  obtain  a 

R0LE  TO   EXTRACT   THE   ROOT,  OF  ANY    DECREE,  OF  A  MONOMIAL. 

1°.  Extract  the  root  of  the  numerical  coefficient  according  to  the  rules  of 
arithmetic. 

2°.  Divide  the  exponent  of  each  letter  by  the  index  of  the  required  root 
Thus, 

J/Gla-'^c6      =4a36c9 

Vl6aft1W=  ' 

•  When  ii  quantity  is  positive,  nil  ll  tivej  bat  it"  it  is  u  D  its 

■ven  powen  will  be  positive,  and  its  uneven  qi      I 


EXTRACTION  OF  ROOTS.  54 

EXAMPLES. 

(1)  {2abc)5z=32a5¥c5. 

(2)  (3a'2m3ni)3=27a6tnPri12. 

(3)  (zmifzr)s=x8mymz8P. 

(^.m+l^n— 3\  7       ^7m+Jy7n— 21 
2n-p+l    /    ==     27u-7p+7     * 

(5)  (m0-432 X p4'23* X  o3,789 X r°,(M |013=m005618p°-66,>42g0-49257ra00<i8. 

/ak\m     akra 

(6)  ^j   =55T. 

p         pji 

(7)  (x^^xi. 

m        pq[ 

(8)  (y»)°»=y»». 

(A4&krcmn\  *     A4^5kp*cmn^ 


V    v 


msn10     2mn2 


pisqio  -  ~  p3q\ 


(12)  i  I™** 
(13)^1 

<14)  x/i 


21q        B2&PZ31 


1042       —      a6/SG 


032^16^24         ©4])2t33 

256        =         2        * 
sin20cos20  i    sin0cos0 


'  tan4i/>  sec6^     tan2i/'  sec3^ 
6  K  3'Q<r-5(«+&)i5(a:+y)-10(&+g— x)20  \  _&ctr-i{a+b)3(x-\-y)-*(b+c—x)* 

When  the  root  to  bo  extracted  is  of  an  uneven  degree,  its  sign  should  be  that 
of  the  given  quantity  ;  when  of  an  even  degree,  it  should  be  i .    (See  last  note.) 

56.  By  the  rule  for  extracting  a  root,  we  perceive  that,  in  order  that  a 
monomial  may  be  a  perfect  power  of  that  degree  whose  root  is  required,  its 
coefficient  must  be  a  perfect  power  of  that  degree,  and  the  exponent  of  each 
letter  must  be  divisible  by  the  index  of  the  root. 

When  the  monomial  whose  root  is  required  is  not  a  perfect  power  of  the  re- 
quired degree,  we  can  only  indicate  the  operation  by  placing  the  radical  sign 
■/  before  the  quantity,  and  writing  within  it  the  index  of  the  root.  Thus, 
if  it  be  required  to  extract  the  cube  root  of  4a-bs,  the  operation  will  be  indi- 
cated by  writing  the  expression, 

y~4aF&>. 
Expressions  of  this  nature  are  called  surds,  or,  irrational  quantities,  or  radi- 
cals of  the  second,  third,  or  n'h  degree,  according  to  the  index  of  the  root  re- 
quired. 

57.  We  can  frequently  simplify  these  quantities  by  the  application  of  the 
following  principle,  which  is  merely  an  extension  of  that  already  proved  in 
(Art.  51). 


56  ALGEBRA. 

The  n"1  root  of  the  product  of  any  number  of  factors  is  equal  to  the  produc 
of  the  n'A  roots  of  the  diffi  rent  factors.     Or,  in  algebraic  language, 

yabcd =  y~a  xVbXVcxVdX • 

Raise  each  of  these  expressions  to  the  power  of  the  degree  n,  then 

{"Veiled )n=abcd 

And, 

(V«x  Vox  VcX  Vd- ")"=(  V«)°X(Vi)nX(  Vc)nX(  W)n 

=abcd . 


Hence,  siuco  the  n'h  powers  of  the  quantities  V abed,  and  y  a  .  \f  b  .  "y/c 
Vd are  equal,  the  quantities  themselves  must  be  equal.  Q.  E.  D. 

This  being  established,  let  us  take  the  expression  y  5ia*b3c",  whose  roct 
can  not  bo  exactly  extracted,  since  54  is  not  a  perfect  cube,  and  the  exponents 
of  a  and  c  aro  not  exactly  divisible  by  3. 

We  have, 

(1)    y'oAaWc-—  V  27  X  2  X  a3  X  a  X  ^  X  c3 


=  V~'7X  Va:jX  tytPx  V2ac* 
by  the  principle  just  proved, 

=  3aby/'2ac2. 
So,  also, 


(2)    V  WaWc*  =  V16  X'3  X  a*  X  ax  b^Xc^X* 

=  vTgxV^x  V&x  V*x  V"sx  V«x  V* 

=2ab"cy3ac2. 


(3)  fi/ld2a'bcli=y6ix3xa6Xa_XbXcu 

=  V64  Xj/jfiX  V<P  X  y/~3  X  VaX  V~b 
=2ac"fy3ab. 

(4)  Vl92=4^3.* 


(5)  5  y 56u* 65=10aiV '? <**>2- 


(G)    V xu>y~6z$m+l  =xay-12ra  ^/z. 
(7)   bS2     ^*i-S3 


*)^/ 


=BbC-s7 — . 


?/l  V     7/1 

In  the  above  expressions,  the  quantities  3ab,  2ab2c,  2aca,  <5cc,  placed  before 
the  radical  sign,  are  called  the  coefficients  of  the  radical. 

58.  There  is  another  principle  which  can  frequently  bo  employed  with  ad- 
vantage in  treating  these  quantities;  this  is, 

Tlie  m"'  power  (fth  <  a  of  any  quantity  is  equal  to  the  nan1*  power  of 

Oiat  quantity.     Or,  in  algebraic  language, 

|a"}m=amn. 


*  A  good  way  of  separating  a  number  into  factors,  some  of  which  arc  perfect  powers,  is 
to  try  perfect  powers  upon  it  as  divisors,  beginning  with  powers  of  the  lowest  nunbaai 
Tims,  in  the  -nil  example,  B,  the  cube  of  9,  will  divide  199,  and  the  quotient  ia  94 1  again,  8 
will  divide  94,  and  the  original  number,  199,  may  be  pot  under  the  form  ^X8X3:=G4X3, 
and  the  cube  tool  will  be  "X-X  ^3,0x4^3.  The  cube  root  of  IPSO  may  be  found  by  first 
tiriding  by  93, and  that  quotient  by  3s, or  97.    The  remit      N  <5=9X3^5==64<5 


EXTRACTION  OF  ROOTS.  57 

For  we  have, 

\as\i=a3Xa:iXa3Xa9 

And,  in  general, 

\an\m^anXanXa"xan---tom  factors; 

/-fn-l-n-f-n  +  n  —  to  m  terms  > 

=am". 
AeJ,  recipi-ocall^, 

The  mn'*  root  of  any  quantity  is  equal  to  the  m"  root  of  the  n*  root  of  that 
quantity.     Or,  in  algebraic  language, 

mn/ —  /  „  r— 

V«=™/  tya. 
For,  let 

^Jya—p  ; 

Raise  the  two  quantities  to  the  power  m, 

y~a=pm  ? 
Again,  raise  both  to  the  power  n, 

a— 2)'""  • 

Extract  the  mn"1  root, 

mn# 

But,  by  supposition, 


41 


yVa=p, 

.-.  mya  =  ^jVa. 

Hence,  as  often  as  the  index  of  the  root  is  a  number  composed  of  two  or 
more  factors,  we  may  obtain  the  root  required  by  extracting,  in  succession 
the  roots  whose  indices  are  the  factors  of  that  number.     Thus, 

(1)  V4^=3*  Vla~", 

=*  h/4a"  by  the  above  principle, 

(2)  V36a2b-=      V36a2b'2 

=  -J  Gab. 

(3)  ^256=4/^256  =  ^/16=2. 

(4)  ^/32oJb5=  y/2ab. 

(5)  ^16a4x2T/2m2-t"-4=  V 4.a?xymz°a-2. 

(6)  In  general, 


myan=™Jyaa 


=  l/«- 
That  is  to  say,  When  the  index  of  the  radical  is  multiplied  by  a  certain 
number  n,  and  the  quantity  under  the  radical  sign  is  an  exact  n'h  power,  we 
can,  without  changing  the  value  of  the  radical,  divide  its  index  by  n,  and  ex 
tract  the  n'/l  root  of  the  quantity  under  the  sign. 


58 

ALGEBRA. 

Thus, 

i/25a*b-c6               =  */oa"bc3, 

ty27  mlWp6            =  ^/3m6n3p'i, 

^/27a:ix3n'y3^-^    =?/3axmyP-*, 

ty  q^^r5"  =  -\Zqj)-lra. 

59.  This  last-  proposition  is  the  converse  of  another  not  less  important, 
wnich  consists  in  this,  that  we  may  multiply  the  index  of  a  radical  by  any  num.' 
ber,  provided  we  raise  the  quantity  under  the  sign  to  the  power  whose  degree  is 
marked  by  that  number,  or,  in  algebraic  language, 

y~a=mya™. 
For,  if  the  last  rule  be  applied  to  the  second  of  these  quantities,  it  will  pro- 
duce the  first. 

60.  By  aid  of  this  last  principle,  we  can  always  reduce  two  or  more  radi- 
cals of  different  degrees  to  others  which  shall  have  the  same  index.  Let  it  be 
required,  for  example,  to  reduce  the  two  radicals  \/2a  and  ^/'3bc  to  others 
which  shall  be  equivalent,  and  have  the  same  index.  If  we  multiply  3,  the 
index  of  the  first,  by  5,  the  index  of  the  second,  and,  at  the  same  time,  raise 
2a  to  the  5th  power;  if,  in  like  manner,  we  multiply  5,  the  index  of  the 
second,  by  3,  the  index  of  the  first,  and,  at  the  same  time,  raise  3bc  to  the  3d 
power,  we  shall  not  change  the  value  of  the  two  radicals,  which  will  thus 
become 

•  ty2a~  =«*^(2a)5  =  ty32a6 


y3bc=3*fy{3bc)3  =  li/27b3c3. 
We  shall  thus  have  the  following  general 

RULE. 

In  order  to  reduce  two  or  more  radicals  to  others  which  shall  be  equivalent 
and  have  the  same  index,  multiply  the  index  of  each  radical  by  the  product  of 
the  indices  of  all  the  others,  and  raise  the  quantity  under  the  sign  to  the  power 
whose  degree  is  marked  by  that  product. 

Thus,  let  it  be  required  to  reduce  -\f2a,  $/3bn-c3,  %/4d*e6f6  to  the  same 
index, 

2/7,7  —3X6X  »/f.;>/7\3X6  —  27 915/715 


y2a       =3X6xy(o(l)8x5        _y 


ty3b-c3     =-x5xs/(36'2c3)2X5      =*y3lob'x>c30 


■v/4d-'e5/6=3X3X  y{id*ef»)-*3=  ^A6d^e30j    - 
The  above  rule,  which  bears  a  great  analogy  to  that  given  for  (he  reduction 
of  fractions  to  a  common  denominator,  is  susceptible  of  the  same  modifications. 

RULE. 

To  reduce  radicals  to  their  least  common  index,  find  the  least  common  multi- 
ple of  all  the  indices,  divide  it  by  the  index  of  each  radical,  and  raise  the 
quantity  under  the  radical  to  thepowt  r  expressed  by  the  quotient.* 
Tins  rulo,  applied  to  the  radicals  {/a.  fybb,  \/2c,  gives 

Va=5v/^'.   VTb=,yG2!Jb\  V 3c=  V'STc3- 
l  \  UUPLES. 

(1)  Reduce  \/am,  \/b",  and  %/<-'  ii>  die  same  index. 

(•J)  Reduce  'y/a,  \/l>,  and  ^/.-  to  the  same  index. 

(3)  Reduce  fyefi,  fyb\  V<\ ;11"1  $/<P  to  the  aame  index.  


*  This  is,  in  effect,  multiplying  the  index  of  cadi  radical,  and  the-  exponents  under  that 
radical.  l»v  the  quotient- 


CALCULUS  OF  RADICALS.  '  59 

(4)  Reduce  W— ,  \J-ri  and  W—  to  the  same  index. 

(5)  Reduce  -J   _,,  ll .  ■,  and  $/-  to  the  same  index 

ANSWERS. 

(1)  >9  a4ra,  '-yi3n,  and  ^c3p. 

(2)  m7"^;  mn^"FF,  and  ran-(/T™. 

(3)  MfyrtW,  ^W,  "tofyfrfi,  and  "Pifd&Br. 

/O       /Dp7  ////.pi 

<4>  "V^  T*r. and  "V^s- 
<5)  v^zs?  VfcFip and  v? 

61.  Let  us  now  proceed  to  execute  upon  radicals  the  fundamental  opera- 
tions of  arithmetic. 

ADDITION  AND  SUBTRACTION  OF  RADICALS. 

Definition. — Radicals  are  said  to  be  similar  when  they  have  the  same  in- 
dex, and  when,  also,  the  quantity  under  the  radical  sign  is  the  same  in  each ; 
thus,  3-\/a,  I2ac-\/a,  156  -\/a,  are  similar  radicals,  as  are,  also,  Aa-b^mn^jP, 
bltymnFjP,  tydynm*]?,  &c. 

This  being  premised,  in  order  to  add  or  subtract  two  similar  radicals  we 
have  the  following 

RULE. 

Add  or  subtract  their  coefficients,  and  place  the  sum  or  difference  as  a  coeffi- 
cient before  the  common  radical.     For  example, 

(1)  3^/bJt-2^T=5^/b. 

(2)  3^6—2^6  =  ^/6. 

(3)  3pq\fim+4ltyrm=:(3pq+4l)tymn.* 

(4)  Sicds/a — 4cd-\Za=5cd-Ja. 

If  the  radicals  are  not  similar,  we  can  only  indicate  the  addition  or  subtrac- 
tion by  interposing  the  signs  -f-  or  — . 

It  frequently  happens  that  two  radicals,  which  do  not  at  first  appeal-  similar, 
may  become  so  by  simplification  ;  thus, 

(5)  V48a63+  6 V75a=  V3Xl6xaX63-f6-v/3X25xa 

=46  V3a-f-56  </3a 
=  96 -/3a. 

(6)  2  V45— 3  V5=2  -/o  X  9— 3  y/Z 

s=3-/5. 


(7)  ^/8a36+16a*—  ^6<+2a63=^8a3(6+2a)  — #6;!(6+2a) 

=  (2a  — 6)^2a+6. 

*  When  two  products,  consisting  each  of  several  factors,  have  any  common  factors,  the 
other  factors  may  be  regarded  as  the  coefficients  of  these,  since  they  show  how  many  times 
the  common  factors  are  repeated,  and  the  addition  may  be  performed  by  adding  the  coeffi- 
cients, and  annexing  the  common  factors  to  the  sum;  thus,  abcd-\-m ncd=iab-\-mn)cd,  and 
5abi/x-{-4cb\/x=(oa-\-4c)b-\/x,  on  the  same  principle  as  8a-\-ia=12a. 


60  ALGEBRA. 

(8)  2VTa*+2y2a=3V2a+2V2a 

=5ty2a. 

(9)  V8+ V50— -/18=4V2. 

(io)  iy ab"+cy~^—dy ad<n={i--\-c"-—(r-)y a. 

(11)  2  V:;+  ^60-  V15+  V$=n  V15-* 

(12)  4a^a3i-»4-i^/ri^6  — ^125a*'6'  =  a-i^/i. 

(13)  V(3a-c-|-  6a6c+3Z/2c)=:(a+5)  -y/3c. 

(14)  v'45c-3—  V«0c3+  v'5a2c=(a— c)  V '5c. 

MULTIPLICATION  AND  DIVISION  OF  RADICALS. 
62.  Li  the  first  place,  with  regard  to  radicals  which  have  the  same  index, 
let  it  bo  inquired  to  multiply  or  divide  y  a  by  yh,  then  we  shall  have 

V'aX  V~b=  Vol,  and   y~a+-  yh=yj"r. 

For,  if  we  raise  y/aX  yh,  an^  V  ah,  each  to  the  7ith  power,  we  obtain  the 
same  result,  ah  ;  hence  these  two  expressions  are  equal.  The  same  principle 
is  demonstrated  in  (57). 

y<i  fa  a 

In  like  manner,  — =  and  \Jr,  when  raised  to  the  nth  power,  give  t  ;   hence 

the  two  expressions  are  equal.     We  shall  thus  have  the  following 

RULE. 

In  order  to  multiply  or  divide  two  radicals  which  have  the  same  index,  mul- 
tiply or  divide  the  quantities  under  the  sign  by  each  other,  and  affect  the  result 
with  the  common  radical  sign.  If  there  he  any  coefficients,  we  commence  by 
multiplying  or  dividing  Oicm  separately.  The  latter  part  of  this  rule  depends 
upon  the  principles  set  forth  and  alluded  to  in  17,  note  ;  the  coefficients,  or  ra- 
tional parts,  and  the  radical  parts  being  regarded  as  factors  composing  a  product 


»)-#^x-3#3S=-c,#!S 


cd 
6a-(a--\-b-) 


Vcd 


(2)  3aV8a-x2hyia-c=Gab  i/32a<c 

=  l2a°bty-2c. 

(3)  2a  y/Tc  X  2b  y/abc  X  a  -/2a = 6a:h  -J-Ja:h:c- 

=  Ga:1i-c-/2. 
5ay/b     5a    K 


(4) 

2by/c      2iVt- 


(5) 


25a-b  Vm3n     o5a^    f~^ 


bah^^mn1       5alr\tnn* 
_ba    Im? 
~b\  n 


■  Tin'  numerator  and  denominator  of  each  of  tin1  two  fraction*  in  tins  example  are  multi- 
plied by  its  denominator.    Tin.'  ilenominator  becomes  thus  n  perfect  square,  ami  may  bo  set 

le  ili''  rad 


CALCULUS  OF  .RADICALS.  gj 


Va3624  b*_9  fobjaW+b*) 

fiCp        V      a2— ¥ 


\f~ 


=26  a@ 


86 

•  62 

(7)  (a+6-/^l)X(«-Z'\/:Zl)=a3+6'. 

(8)  '/flXyixV^yaic. 

(9)  atyxxWyX^Vz=ateyxyz. 

(10)  4X^V~3X  V72=8V6. 

(11)  c*\/axd^a=acd. 

(12)  5-v/8x3vf5=30V,10. 

(13)  ^18X5^4  =  10^9. 

(14)  »V6XT33/9f=TVV6.  

(15)  2  ^  X  3  Va$  X  4  ^6<5=24  V^+^R 


(17)   (V-15+V-12-V-21)4-V-3  =  2+  V5-     -v/7- 
If  the  radicals  have  not  the  same  index,  we  must  reduce  them  to  others 
having  the  same  index,  and  then  operate  upon  them  as  above ;  thus, 

(1)  2aylxrob\/2c=Za^'biX^h^/S^ 
=15abfy8b*c?. 


(2)    jbab<?x  V 2a*bc*—  yvZbcP&d* X  tylaWc* 

=  V500a765c13 


=ac2V500a65c. 
(3)  my  a  X  y/b  x  n  %/c=mn  ty  a15620c12. 

a    jc     x    jz     ax    IcV 
^  by]dXy\u~by^d^' 

(5)  x$/ms  X  y^tm=xy  ym^i*. 

(6)  V^T^X  ^"-6xV(a*-6*)»=V~^=6T- 

(7)  VaX  V^X  Vc=m7anP6mPcmn. 

(8)  AV«'r'XB'{/i"XCVcr=ABCaV«m^Da''cr^- 

/ami  /am-JC3  /am(3n-2)+i2j3n^l0-3n 

(10)  c  y/a?—x*-±-  1/a+x=cV(a— xf(a"— x2). 

(11)  ^Z?4.(a-z)=^E.  | 

Ira  <rv        A  l,.an— im      >^ 

(12)  A,™  -  +  A„J--^Jx--z-r 

\yp        ~\'ys     A2~<JyPa-°m 
FORMATION  OF  POWERS  AND  EXTRACTION  OF  ROOTS  OF  RADICAL8. 
63.  Let  it  be  required  to  raise  tya  to  the  nth  power;  then, 

(Vajn=  yfaX  V«X  ^a to n "factors, 

=  tyan,  according  to  the   rule  for  multiplication  just   established. 
Hence  we  have  the  following 


62  ALGEBRA. 

RULE. 

In  order  to  raise  a  radical  quantity  to  any  gicen  powet    raise  the  quantity 
under  the  sign  to  that  power,  and  place  over  the  result  the  radical  sign  wiOi  its 
original  index.     If  there  he  any  coefficient,  we  must  raise  die  coefficient  sepa 
rately  to  the  required  power.     Thus, 

(1)  {\/Ta?y=  Vl6a6 
=2a  \TcP. 


(2)  (3V2a)5=35V32a6 


=243-^  32a5 
3=486aV4a». 
When  the  index  of  the  radical  is  a  multiple  of  the  exponent  of  the  power 
which  we  wish  to  form,  the  operation  may  be  simplified. 

Let  it  be  required,  for  example,  to  square  ty2a ;  we  have  seen  (Art.  58)  that 

^/2a=yJ  V~a  i  Dut  m  order  to  square  this  quantity,  it  is  sufficient  to  sup- 
press the  first  radical  sign ;  hence,  ( y/2a)-=  <J '2a.     Again,  let  it  be  required 

to  raise  y/abc  to  the  5th  power;  now,  1tyabc=\!  ^abc;  but  in  order  to  raise 
this  quantity  to  the  5th  power,  it  is  sufficient  to  suppress  the  first  radical  sign  • 
hence,  {^abcf=z  -J  abc,  and,  in  general, 

(^)ra=W  W  =V~a; 
that  is  to  say, 

If  the  index  of  the  radical  be  divisible  by  the  index  of  the  required  power,  we 
may  divide  the  index  of  the  radical  by  Oie  index  of  the  power,  and  leave  the 
quantity  under  the  sign  unchanged.* 

64.  With  regard  to  the  extraction  of  roots,  either  by  virtue  of  the  principle 
established  in  (Art.  59),  or  by  reversing  the  last  rule,  we  shall  manifestly  have 
the  following 

RULE. 

In  order  to  extract  any  root  of  a  radical  quantity,  multiply  Oie  index  of  the 
radical  by  the  index  of  the  root  required,  and  leave  the  quantity  under  the  sign 
unchanged.  If  there  be  a  coefficient,  we  must  extract  its  root  separately. 
Thus, 


(1)  V i/Tc=iy'3c~. 

(2)  V  V5«"=  V5a\ 

(3)  yj8<?\/arb=2cy~l 


If  the  quantity  under  tho  sign  bo  a  perfect  power  of  the  samo  degree  as  the 

root  required,  we  may  simplify.     Thus, 

— — . — — , *  M 

•  It  may  !»■  w  ell  to  noto  hero  thai  the  b\  en  pon  er  of  a  radios]  of  the  second  degree  is 
rational,  and  the  uneven  power  Irrational,  the  tatter  being  farmed  by  the  multiplication  of 
tho  proponed  radical  by  a  rational  quantity. 


CALCULUS  OF  RADICALS-  63 

=  yza ; 
that  is,  we  may  extract  the  root  of  the  quantity  under  the  radical  sign. 

MISCELLANEOUS  EXAMPLES. 

V24+  V54—  \/6=4  VG- 
V"l2+2  V27  +  3  V^5+9  V48=59  y/5. 
V81  — 2  V24+  V28+2  V63=8  </7  —  ¥$- 
V45c3—  Vr80c3+  ■)/~baTc'={a— c)  </bc. 


( 


Vl8a563+  V50a363=(3a26+5a6)  V2a6. 


V 2Ha1365c—  V4  X  54a569c5+  4/4  X  64a55c  

=  (8a36  — 5a62c  +  66)y4a6c. 

8  f27aBx_3  g£=(3a_i) 
V~2T     V26 


/y2i 


26 


^54ara+663— ^/16am-366+ ^2a4m+9+ ^/2c3am 

262  

=  (3a26  — —  +  am+3+c)^/2am. 


^3X2¥/4     #23^  f  /      #2   j     /3X2V^ 


^^  y35c3^/2     < «    3crf  >  v    /2g 


d^3cd\-\/~ 


*M 


8a*      16a3\      2a.r 


m3+2762i=^^+26- 
V"4"a32/+8a6i/+462?/=2(a4-6)  ■/#■ 
A/4a562— 20a363+25a&4=(2a2— 56)6  >' al 

•v/a2x — 2ax2-f-^'3     a — x    ^_ 

y/a2+2ax-\-x'i       a-\-x 

a — 6  -\f  ac  -\fac 

a+b"  1/a2_2a6+P~ «+&' 

a-j-6     /a — 6        /a+6 
^6-V^+6==V^36' 

V2x\|xV3=2</^. 

V4  X  7~3  X  ^6=  ^3981312 

atyxxb  y/y  X  c  \Tz=ahc  mnty  xBPymPzma. 

\a™     2  Ja3m+2 
~b~~\  65c9  ' 


"£* 


It  is  manifest  that,  in  gnneral,  V  ■£/<*=  Vv/a;  for,  by  (Ait.  58),  each  of  these  expre# 
eions  is  =  v  '«• 


C4  ALGEBHa. 

,o0)     <"3lM      Wtt       l.&V. 

65.  Let  us  now  inquire  with  what  sign  a  moncmial  root  is  to  be  affected. 
We  have  seen  (Art.  52)  that,  whatever  may  be  the  sign  of  a  monomial, 

its  square  is  always  positive ;  and  it  is  evident  that,  in  like  manner,  every  even 
power  must  be  positive,  whatever  may  bo  the  sign  of  the  original  monomial, 
and  that  every  uneven  power  will  be  affected  with  the  same  sign  as  the  origina 
monomial. 

Thus,  — a,  when  raised  to  different  powers  in  succession  will  give 

—a,  -{-a2,  — a'>  +  «'S  — <*5»  +q6>  — a7i  &c. 
A.nd  -\-a,  in  like  manner,  will  give 

-\-a,  +a2,  -{-a3,  -\-a\  -\-a\  +  a<5»  +a7,  &c. 
In  fact,  every  even  power  2n  may  be  considered  as  the  square  of  the  ?ith  powei 
or  a2n  =  (an)2,  and  must,  therefore,  be  positive;  and,  in  like  manner,  every 
power  of  an  uneven  degree  (2«-|-l)  may  be  considered  as  the  product  of  the 
2«tU  power  by  the  original  monomial,  and  must,  therefore,  have  the  same  sign 
with  the  monomial. 
Hence  it  appears, 

I.  Tliat  every  root  of  an  uneven  degree  of  a  monomial  quantity  must  be 
iffected  with  the  same  sign  as  the  quantity  itself.     Thus, 

V  +  8a3=2a;  y  —  8a3  =  —  2a;  V~ 32a1065=  —  2a-b. 

II.  That  every  root  of  an  even  degree  of  a  positive  monomial  may  be  affected 
with  the  sign  +,  or  (he  sign  — ,  indifferently.     Thus, 

V81a<6:-=±3a&3;  ^/6Aals=±2a3. 

III.  That  every  root  of  an  even  degree  of  a  negative  monomial  is  an  impos- 
sible root ;  for  no  quantity  can  be  found  which,  when  raised  to  an  even  power- 
can  give  a  negative  result.     Thus,   V — a»  V — c, . . .  are  symbols  of  opera 
tions  which  can  not  be  performed,  and  are  called  impossible,  or  imaginary 
quantities,  as  V — a,  •/  —  b,  in  (Art.  53). 

66.  The  different  rales  which  have  been  established  for  the  calculation  oi 
radicals  are  exact  so  long  as  we  treat  of  absolute  numbers;  but  are  subject  to 
some  modifications  when  we  consider  expressions  or  symbols  which  are 
purelytalgebraical,  such  as  the  imaginary  expressions  just  mentioned. 

Let  it  be  required,  for  example,  to  determine  the  product  of  1/ — a  by 
■/  —  a  ;  by  the  rule  given  in  (Art.  62), 


V—  «X  V  —  a=  V—aX  —a 

=  V  +  cr. 


But  V+":=iai  so  that  there  is  apparently  a  doubt  as  to  the  sign  with 
which  a  ought  to  bo  affected  in  order  to  answer  th>'  question.  However,  the 
true  result  is  — a  ;  because,  in  general,  in  order  bo  Bqnare  •/>/>,  it  is  sufficient 
to  suppress  the  radical  sign ;  but  \t — <tx  V — "  is  the  same  thing  as  (  ^ — </)* 

and,  consequently,  is  equal  to  — a- 


Next,  let  it  be  required  t"  determine  the  product  of  -y/ — a  by  y*~l>;  by 

the  rale  (Art.  62)  

V^-aX  ■/  —  ''=  V—aX—b 
=  V  +  «6 


Pot  farther  i-xaniples  of  traoifbrmatiooa,  sec  Appendix. 


FRACTIONAL  AND  NEGATES  EXPONENTS.  6fc 

The  true  result,  however,  is  —  V 'ab,  so  long  as  we  suppose  the  radicals 
y/  —a,  V—  &  to  be  each  preceded  by  the  sign  -j-  ?  for  we  have,  according 
to  (Art.  53), 


Hence, 


V  -a  X  V  -b~  Vab(  V  -I)2 
=  \/«^X  —1 
=  —  -\/ab. 
According  to  this  principle,  we  shall  find  for  the  different  powers  of  ■/  —  1 
the  following  results : 


& 


V-l  =V-i 

(v-ir=-i 


(V-i)*=(V-i)2x(V-i)2 

=-ix-i 
=+i. 

Since  the  four  following  powers  will  be  found  by  multiplying  +1  by  the 
first,  the  second,  the  third,  and  the  fourth,  we  shall  again  find  for  the  four  new 
powers  -f-  -/ — 1,  — 1,  —  •/  —  1,  -f-1  ;  so  that  all  the  powers  of  -y/ — 1  will 
form  a  repeating  cycle  of  four  terms,  being  successively,  -/  —  1,  — 1,  —  ■/  —  1, 

-H-*  

Finally,  let  it  be  required  to  determine  the  product  of  4/ — a  by  </ — b, 
which,  according  to  the  rule,  would  be  ty-\-ab.  To  determine  the  true  result. 
we  must  observe  that 


V— a  = V a  .  V  — 1 

V—  b  =V6.V  — 1- 


And  .-. 
But, 

Hence, 


=  y-i. 


y—aX  \/—b=yab.j—\. 
The  above  principles  will  enable  the  student  to  operate  upon  these  quanti- 
ties without  embarrassment. 

THEORY  OF  FRACTIONAL  AND  NEGATIVE  EXPONENTS. 
67.  This  is  the  proper  place  to  explain  a  species  of  notation  which  is  found 
extremely  useful  in  algebraic  calculations. 

*  This  may  be  expressed  in  its  most  general  form  thus,  if  n  be  any  whole  number  : 
(a\f—i.)*a     =a*nX+l  =a*° 

(aV^)4a+l=a-«n+lX-fV---1=:a4D+l .  V~i 

(a\/— l)-»°+2=a<n+2X— 1         =— a<n+2 

Tbe  first  in  the  note  corresponds  to  the  last  in  the  text,  the  second  in  the  note  to  the  first 
in  the  text,  and  the  third  in  the  note  to  the  second  in  the  text. 

E 

» 


66  ALGEBKA. 

]  Let  it  be  required  to  extract  the  71th  root  of  a  quantity  such  as  am.  We 
have  seen  by  (Art.  55)  that,  if  m  is  a  multiple  of  n,  wo  must  divide  m,  the 
index  of  the  power,  by  n,  the  index  of  the  root  required.  But  if  m  is  not 
divisible  by  n,  in  which  case  the  extraction  of  the  root  is  algebraically  impos- 
sible, we  may  agreo  to  indicate  that  operation  by  indicating  the  division  of  the 
exponents.     We  shall  thus  have 

m 

ya™=a°, 

m 

the  expression  aa  being  understood  to  signify  the  71th  root  of  am,  by  a  conven 
tion  founded  upon  the  rulo  for  the  extraction  of  roots  of  monomial  quantities. 
According  to  this  convention  or  definition,  wo  shall  have 

tya-=J;  !/iv=aJ. 

It  may  be  observed  that  the  denominator  of  the  fractional  exponent  is  the 
index  of  the  radical,  and  the  numerator  the  exponent  of  the  quantity  under  the 
radical. 

II.  Let  it  be  required  to  divide  am  by  an.  According  to  the  rule  in  (Art. 
17),  we  must  subtract  the  index  of  the  divisor  from  the  index  of  the  dividend ; 
so  that 

am 

— z=am~° ; 
aa 

it  is  to  be  remarked,  however,  that  here  it  is  supposed  that  m  >  n.    But  if 

m  <  n,  in  which  case  the  division  is  algebraically  impossible,  we  may  agree  to 

indicate  the  division  by  the  aid  of  a  nogativo  index  equal  to  the  excess  of  n 

over  m.     Let p  be  the  absolute  difference  of  m  and  n,  so  that  n=m-{-p  ;  we 

shall  then  have 

am       am 


__flm— (m+p) 

=arr. 

am  1 

But  -^jr  may  also  be  put  under  tho  form  — ,  by  suppressing  the  factor  aa 

common  to  both  terms  of  the  fraction ;  we  shall  then  have 

1 

o-p=— . 
a? 

The  expression  or?  is  then  tho  symbol  of  a  division  which  can  not  be  exocuted ; 
and  tho  true  value  of  the  expression  is  unity  divided  by  the  same  U-tter  a 
affected  with  tho  exponont^>,  takon  positively.  According  to  this  convention, 
we  shall  have 

3        l  -»        1     At 

a~3=:— :  a  ■=— ,  <Vc. 

Again,  by  supposing  the  exponent  of  the  numerator  to  be  larger  by  j>  than 
tho  exponent  of  the  denominator,  it  may  bo  proved  in  B  similar  maunor  that 

1 
aP=— . 

a   i 

From  these  expressions  it  appears  that  a  factor  may  bo  transferred  from  the 
denominator  to  the  numerator  of  a  fraction,  or  vice  versa,  by  changing  tbo  pi^n 
of  its  exponent 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.  67 

EXAMPLES. 


aW 


Write  -ttj  in  one  line.  Ans.  a26*c_sd"^. 

2amCa 

Write  —5 —  in  one  line.  Ans.  3amcad~fe'  <*. 

dve^ 

Write  -    -  is  m  one  ^ae'  ■^■ns'  2  x  3_Ign7i'»-r&*. 

a564  1 

Write  -rr„  all  in  the  lower  line.  Ans. 


A«B^C  1 

Write  — tt —  all  in  the  lower  lino.  Ans 


A5C-4  A6B6 

Write  v;_6r>3  with  all  positive  exponents.  Ans.  T^ffr 

aab—$  aad£ 

Write    y,_5  with  all  positive  exponents.  Ans.  -zttj- 

III.  By  combining  the  last  two  conventions,  we  arrive  at  a  third  notation, 
which  is  the  negative  and  fractional  exponent. 

Let  it  be  required  to  extract  the  nlh  root  of  — . 

1  PT        _J2 

In  the  first  place,  — =a~m ;  hence  y—=y/a~m=a   n>  substituting  the 

fractional  exponent  for  the  ordinary  sign  of  the  radical. 

As  in  words,  am  is  usually  enunciated  a  to  the  poioer  m,  m  being  a  positive 

m  m 

integer ;  so  by  analogy,  a",  a~m,  a  n  are  usually  enunciated,  a  to  the  power  m 
by  n,  a  to  the  power  minus  m,  and  a  to  the  power  minus  m  by  n. 

All  that  has  been  hitherto  said,  with  regard  to  fractional  and  negative  ex 
ponents  must  be  considered  as  a  mere  matter  of  definition ;  in  short,  that  by  a 

m 

convention  among  algebraists  aa  is  understood  to  mean  the  same  thing  as 

1  _iH  Jl 

V  am,  a~m  to  be  the  same  as  — ,  and  a   n  as  a  /— .     We  shall  now  proceed  to 

prove  that  the  rules  already  established  for  the  multiplication,  division,  forma- 
tion of  powers,  and  extraction  of  roots  of  quantities  affected  with  positive  in- 
tegral exponents,  are  applicable  without  any  modification,  when  the  exponents 
are  fractional  or  negative.     We  shall  examine  the  different  cases  in  succession. 

3  j| 

68.  Multiplication.     Lot  it  bo  required  to  multiply  aJ  by  a*  ;  then  it  ia 
asserted  that  it  will  be  sufficient  to  add  the  two  exponents  and  that 

3  2  3    I     3 


19 

For,  by  our  definition, 

3 

a* 

=  y~a~\ 

And, 

2 

a? 

=  vV; 

.•.  a5Xa^=V«3X  Va' 


S3  V«19 

=aT3  by  definition  in  (Art.  67,  I.), 


orf  ALGEBRA. 

3  s 

Again,  lot  it  be  required  to  multiply  a     *  by  a7 ;  then  it  is  asserted  mat 

3        5         _  s  •  «  \    . 

8      110 


=a' 


For, 


_3     /r       &     

a      f=\J—,  and  a*=  V«5 

_3         *      A  fl  

•.a     *xa«=\J-3xVa* 


i 

=a12  by  definition  in  (Art.  67,  1.) 

m  p 

Generally,  let  it  be  required  to  multiply  a     n  by  a*  ;  ther 


p  «« j  p 


a     nxaq=a     n     q 


np — mq 


=a  ni 
For, 


i   T=V5= 


p         

and  a^=  VflP 


m  p 


np— mq 

=a   ni     by  definition. 
69.  Hence  we  have  the  following  general 

RULE  FOR  EXPONENTS  IN  MULTIPLICATION. 

In  order  to  multiply  quantities  expressed  by  the  same  letter,  add  the  ex 
ponents  of  that  letter,  whatever  may  be  the  nature  of  the  exponents. 

This  is  the  same  rule  as  was  established  in  (Art.  11)  for  quantities  affected 
with  integral  and  positive  exponents.     According  to  this  rule,  we  shall  find 

3       _3  ,  2    3  It     5    2 

afb     2c        xa-Pcs=aib     *c     ■ 
3a-*b* X  2cT~h*&  =6a~  T  iV. 


i 


70.  Division.  Let  it  be  required  to  divido  a-  by  aT;  then  it  is  asserted 
that  it  will  bo  sufficient  to  subtract  the  index  of  tho  divisor  from  the  index  of 
the  dividend,  and  that  we  shall  thus  have 


a-       ?,-\ 
sra*. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.  69 

For, 

3  1  _ 

a2  =  "y/a*,  and  a*  =  \/c, 

3  

.•.  a2       V a? 


=V-  by  (Art.  62, 
=  V^ 

5 

=a4  by  definition. 
In  like  manner,  we  can  prove  that 


o 
<*T  5 (_3\ 


_3—a 
a     * 

13 

m  p 

Generally,  let  it  bo  required  to  divide  aa  by  a"). 
Then, 

m  p  m        p 

mq — np 


For, 


m  p 

an=  V"a™,  and  a^=  V"cp, 
m       p      V"a" 

■•■a"+a*=vaT 

Va"P 


mq— np 

=a    "i     by  definition. 

71.  Hence  we  have  the  following  general 

RULE  FOR  EXPONENTS  IN   DIVISION. 

In  order  to  divide  quantities  expressed  by  the  same  letter,  subtrac  the  ex- 
wonent  of  the  divisor  from  the  exponent  of  the  dividend,  whatever  may  be  the 
nature  of  the  exponents. 

This  is  the  same  rule  as  that  established  in  (Art.  17)  for  quantities  affected 
with  integral  and  positive  exponents.     According  to  this  rule,  we  have 

2  __3  2 /         3\ 

a3  +  a     4=a3     *     *' 

17 

=a12. 

3  4  1 

aT-i-a~s   =a     20. 

2    3  1     7  9  1 

aH*+a     sb*=a10b     *. 

72.  Formation  of  powers In  order  to  raise  a  monomial  to  any  power, 

the  rule  given  in  the  case  of  positive  and  integral  exponents  was,  to  multiply 
the  index  of  the  quantity  by  the  index  of  the  power  sought.  We  have  now 
to  prove  that  this  holds  good,  whatever  may  be  the  nature  of  the  exponent. 


70  ALGEBRA. 

5 

Let  it  be  required  to  raise  a1  to  the  4lb  power. 
Then, 

(a1)  =a^ 


For, 
But, 


20 

=a  * . 


J=  Va\  and  (o"V  =  (  Va6)4- 


(  V«5)4=  Va20,  by  (Art.  63) 

20 


Generally,  let  it  be  required  to  raise  an  to  the  power  p. 
Then, 


(#=a> 


Dip 

=a~. 


For, 
But, 


m  /    m\p 

a°  =  V~cF,  and  \o°/  =.(  V"a™)r 


( V«ra)p=  Vamp 

rap 


The  demonstration  will  manifestly  be  precisely  the  same  if  we  suppose  one 
or  both  of  the  indices  to  be  negative. 

73.  Hence  we  have  the  following  general 

RULE  FOR  RAISING  A  MONOMIAL  TO  ANT   POWER. 

Multiply  the  exponent  of  the  monomial  by  the  exponent  of  the  power  required, 
whatever  may  be  the  nature  of  the  exponents. 

This  is  the  same  rule  as  that  established  in  (Art.  55)  for  quantities  affected 
with  positive  integral  exponents.     According  to  this  rulo,  we  have 

•  (a4)  =a*X 


15 

=  0 

l-    :: 


=a' 


(a*)  =ac'X3 


:i    '  .fX8 


=a 

(2a~~^>4)  =2ba_-^A1 

=64a       ' 

74.  Extraction  of  Roots. — Tn  order  to  extract  the  «"'  root  of  any  quan- 
tity nccordiug  to  the  rulo  in  (Art.  55),  we  must  divide  the  exponent  of  each 
letter  by  tho  index  n  of  tho  root.  Let  us  examine  the  case  of  fractional  ex- 
oononts. 

Lot  it  bo  required  to  extract  tho  cube  root  of  a'-'. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.  71 

Then, 


=a*. 


For, 


But, 


5  3/53/      

a3=  Va\  and  .-.  V«3=V  V<* 


3rz 

V  V"5 


'  ya?=  Va\ 


=a°,  by  definition. 

m 

Generally,  let  it  bft  required  to  extract  the  jp,h  root  of  a". 
Then, 


^ 


ra  m 


For, 
But, 


aa=\am,  and  .•.yaa=  y  tyam. 


V,«ra=nv/am»  (by  Art.  58), 

m 

=a"P,  by  definition. 
75.  Hence  we  have  the  following 

RULE  FOR  THE  EXTRACTION  OF  ANT  ROOT  OF  AN  ALGEBRAIC   MONOMIAL. 

Divide  the  exponent  of  the  monomial  by  the  exponent  of  the  root  required*, 


whatever 

may 

be  the  nature  of  the  exponents.     Thu9, 

3/  _!       _^3 

\Ja     °=a     5 

0 

=a~~TX 

■\Ja1b~2=a^~1b 

3         2 
=a3jb     *. 

76.  We  shall  close  this  discussion  by  an  operation  which  includes  the  demon, 
stration  of  eveiy  possible  variety  of  the  two  preceding  rules. 

m  y 

Let  it  be  required  to  raise  a"  to  the  power  of  — ;  we  must  prove  that 

s 

m  r  m  r 

(a°)— *=o°      ~~7 

mr  \ 
n         "3 


J2  ALGEBRA. 

If  we  recur  to  the  origin  of  this  notation,  wo  find  that 


m  r        .  / 

_.n~ 

V(Vam)r 


V  \Jamr 

x  amt 


mr 

=a      "",  by  definition. 
77.  The  notation  above  explained  can  be  extended  to  polynomials,  by  in- 
cluding them  within  brackets,  in  the  same  manner  as  was  explained  in  the  case 
of  integral  exponents. 


Thus,  (x-f-a)2  signifies  the  same  thing  as  ■y/x-\-a,  or  tlie  square  root  of 

x-\-a. 

i  1 

So,  (x+fi)      2  is  equivalent  to  ,  or  unity  divided  by  the  square  rovl 

■y/x-\-a 

ofx-\-a. 


In  like  manner,  (:r-|-a-|-o)*  will  be  the  samo  as  ty{r-\-a-\-b)*,  or  the  fourth 

3 

root  of  the  third  poicer  of  the  quantity  x-\-a-\-b,  and  (x-j-a-r-^)     4  will  be 

anity  divided  by  the  last-mentioned  quantity.     Since  unity  is  always  under- 

itood  to  be  the  exponent  when  no  other  is  expressed,  (x-\-a)~l  is  the  same  as 

— ; — ,  and  so  on.     The  samo  rules  which  have  been  established  for  the  treat 
x-|-« 

ment  of  monomials  affected  with  exponents  will  also  manifestly  apply  to  poly- 
nomials under  the  samo  restrictions.* 

EXAM  11 
_3  7  13  1 

(I)  a     *xa     *=a       s  =—=. 


4,  J-  '     — 3      c.nla 


a  y& 

(2)  a~*b~  " Xatb*c=arsb~*K=:£y/ 

7 

,  .      o.         a^b  iS   1        ,  1-7— 

(3)— x— =</»>',-!=   s/Z 


&V  C       "2 


,r\  ,ir- 


*  The  calculus  of  fractional  exponents,  says  Lacroix,  is  one  of  the  most  remarkable 
unpies  of  the  utility  of  signs,  when  they  are  wi  a.    'J"  h .  ■  analogy  which  eacists  be- 

tween traotiona]  and  entire  exponents  renders  the  roles  to  be  followed  in  the  calculus  of 
the  1  ii,  r  applicable  to  the  former,  while  particular  roles  are  requisite  for  the  calculus  of 
'-'.    The  farther  we  advance  in  al  cobra,  the  more  we  perceive  the  nnmerooa  a<lv:m- 
which  have  resnlted  to  that  science  tram  Uic  Dotation  of  exponents   Invented  ly 
Descartes. 


(4 
(5 

(G 
(7 

(8 
(9 

(10 

(11 
(12 
(13 

(14 

(15 

(16 
(17 

lie 

(19 
(20 


FRACTIONAL  AND  NEGATIVE  EXPONENTS.  73 

m  p  p     m  np — mq 

a     a~a     ^=ai  n=a    "i 

3  5       c I 

ca7i^rdo?=2-a      12' 

3 
3     1  7     1  „27,T 


_n     3  _29     II  11     34 

a 263  ,  a      *  d3  _a*bl11 

c*d3  65  c*<2 ' 

(3     2\J.  12 

a*b3)3=a7(b5. 

(I      _2\_1  1     1      I 

a*b     2c     5j     *=a     26V° 

S      c"d      f       3      c     3d     3 


<(a+6)^  (a +  6)      3 

(1  112  1    4  S\        /   JL  1\ 

a'3'+as63+a86T+a6+o263  +  fc3)  X  (a2—  i3)=a3— 4». 

(1        l    i         i\       /   '         U         3         3 
xa+x*y*+ys)  x  VcT— y*J  — ^  — 2/4- 

(.r2+7/2)X(.r     2  +  t/     •)=««,     2  +  2+x     22/2. 
a3— b3 

9  3    3  3    3  O 

|  3=aT_a^4+a462_&4> 


=a2— 6. 


1  2 

a3— b     3 

(3  3\         /     1  1\  I  1  I 

B*.— &*)  :  (a4— Z/4)=a2  +  &2  +  (aZ>)<. 

3      5    1  13         »  99      3_3_18    27     13 

m?piq2riXP      'Vr     4m13X.p32?4=m  *.P  105?  2  r  *  • 

1      Sfl 

2    3  2  17 

a2bTc-5d3^-aib2csa-8= 


a*d3 

5     37 

4 


(z'  +  6Z'a*  +  9a^)  .  -/Z&8 .  (7i/Z  +  3  Vfls)  =  (z*+3a;)3 -i  V5*8 


It  may  be  asked  here  whether  the  rules  for  the  calculus  of  exponents  apply  to  incom- 
mensurable and  imaginary  exponents. 

With  regard  to  incommensurable  exponents,  it  may  be  said  that  they  have  not  absolutely 
of  themselves  any  signification,  and  that,  in  order  to  give  them  one,  it  is  necessary  to  con 
ceive  them  in  imagination,  replaced  by  their  approximate  commensurable  values.  A  forrnu 
la,  therefore,  into  which  incommensurable  exponents  enter,  should  be  considered  as  repre- 
senting the  limit  toward  which  the  values  deduced  from  it  tend  by  the  substitution  of 
commensurable  numbers  for  the  exponents,  differing  from  them  by  as  small  a  quantity  as 
we  choose  to  assign ;  in  this  way  we  perceive  that  the  proposed  expression  will  represent 
exactly  this  same  limit,  when  the  same  operations  shall  have  been  executed  upon  the  in- 
commensurable exponents  which  it  contains,  as  would  be  if  they  were  commensurable. 

Thus,  for  example,  m  and  n  being  incommensurable  quantities,  we  shall  always  have 

amX«n=atn+n- 
For,  if  ir.'  and  nf  represent  their  approximate  commensurable  values,  we  have 

am'  X  an'=am' + "'. 

*  For  a  variety  of  examples  in  transformations,  see  Appendix. 


74  ALGEBRA. 

The  first  members  of  this  equality  tend  toward  the  same  limit  as  the  second.  But 
amXan  represents  the  limit  of  the  one,  and  am+"  that  of  the  other;  hence,  amXan=am+n 

With  regard  to  imaginary  exponents,  there  is  necessary  here,  as  every  where,  a  tacit 
admission  that  the  general  relations  of  real  quantities,  represented  by  letters,  hold  good  when 
these  letters  are  replaced  by  symbol  titiea  which  are  imaginary. 

This  subject  will  be  better  understood  after  the  student  has  been  over  that  of  extrac- 
tion of  roots  by  approximation. 

78.  Having  thus  discussed  the  formation  of  powers,  and  the  extraction  of 
roots  in  monomial  quantities,  wo  shall  now  direct  our  attention  to  polynomials  ; 
and,  in  the  first  place,  let  it  be  required  to  determine  the  square  of  x-\-a  , 
then, 

(x+a)*=(x+a)x(x+a) 

=x°-\-2xa-\-a-  by  rules  of  multiplication. 

By  inspection  of  this  result,  it  is  perceived  that  the  square  of  a  binomial  con 
tains  the  square  of  each  term  together  with  twice  the  product  of  the  two. 

Next,  let  it  be  required  to  form  the  square  of  a  trinomial  (x-\-a-\-b).     Let 
us  represent,  for  a  moment,  the  two  terms,  x-\-a,  by  the  single  letter  z 
Then, 

(x+a+by=(z-\-by 

=z*+2zb  +  b'i (1). 


But, 

And, 


z"={x-\-ay 
=x*+2xa  +  a": 


2zb=2b{x+a) 
=2xb-\-2ab. 
Therefore,  substituting  for  z"  and  2zb  their  values  iu  (1),  we  find 
(x+a  +  b)-=x--\-a'i+b~+2xa+2xb  +  2ab. 

Hence  it  appears  that  the  square  of  a  trinomial  is  composed  of  the  sum  of  die 
squares  of  all  the  terms,  together  with  the  sum  of  twice  the  products  of  all  Oie 
terms  multiplied  together  two  and  two. 

We  shall  now  prove  that  this  law  of  formation  extends  to  all  polynomials, 
whatever  may  be  the  number  of  terms.  In  order  to  demonstrate  this,  let  us 
suppose  that  it  is  true  for  a  polynomial  consisting  ^>f  n  terms,  and  then  en- 
deavor to  ascertain  whether  it  will  hold  good  for  a  polynomial  composed  of 
(n-\-\)  terms. 

Let  x-\-a^-b-\-c-\ ML"+^  D0  a  polynomial  consisting  of  n-f-1  terms, 

and  let  us  represent  the  sum  of  the  first  n  terms  by  the  single  letter  Z  ;  then 

(*+«-£&+«+.—  +*+*)  =(*+*), 

and  .•■.(i+a+&+c4----  +  /j+/);  =  (r  +  0i 

or.  putting  for  z  its  value,  =(x-f-a-f-6+eH \-k)'*  +  2(x+a  +  0 

+  c+ --+/)/  +  /--. 

But  the  first  part  of  this  expression,  being  the  square  of  a  polynomial  con 
sisting  of  n  terms,  is,  by  hypothesis,  composed  of  the  sum  (4'  the  squares  of 
all  the  terms,  together  with  twice  the  sum  of  the  products  of  all  the  tonus 
multiplied  two  and  iw<>;  the  Becond  pari  of  the  above  expression  is  equal  to 
twice  thi  fthe  products  of  all  the  first  n  terms  of  the  proposed  poly- 

nomial, multiplied  by  the  (/t-f-l)'''  term  /;  and  tin'  third  part  is  the  square  of 
tho  (n+1 )    term  I 


SaUARE  OF  A  POLYNOMIAL.  75 

Hence,  if  the  law  of  formation  already  enounced  holds  gcod  foi  a  poly- 
nomial composed  of  n  terms,  it  will  hold  good  for  a  polynomial  composed  of 
{n-\-\)  terms. 

But  we  have  seen  above  that  it  does  hold  good  for  a  polynomial  composed 
of  three  terms  ;  therefore  it  must  hold  for  a  polynomial  composed  of  four  terms, 
and  therefore  for  a  polynomial  of  five  terms,  and  so  on  in  succession.  There- 
fore the  law  is  general,  and  we  have  the  following 

RULE  FOR  THE  FORMATION  OF  THE  SQUARE  OF  A  POLYNOMIAL. 

The  square  of  any  'polynomial  is  composed  of  the  sum  of  the  squares  of  all 
the  terms,  together  loith  twice  the  sum  of  the  products  of  all  the  terms  multiplied 
together  two  and  two.     According  to  this  rule,  we  shall  have, 

(1)  (a  +  6  +  c+rf+e)s=as+68+cs+d3+c9+ 2ab  +  2ac+2ad+2ae+2bc 
+2bd+2be-\-2cd-\-2ce+2de. 

(2)  (a  —  b— c+dy=a?+b*-\-c'2-\-d~— 2ab— 2ac-\- 2ad-\-2bc— 2bd— 2cd. 
If  any  of  the  terms  of  the  proposed  polynomial  be  affected  with  exponents 

~>v  coefficients,  we  must  square  these  monomials  according  to  the  rules  already 
established. 

(3)  (2a— 4Z>2c3)2=4a-s4-16Z>4c6— 16aZ>2c3. 

(4)  (3a*_ 2a6+462)2=9a4+4a262-f  1664 — 12a36 

+24a2l2— 16ab3 
=9a*  —  12a3Z>+ 28a2i3  —  16aZ>3  -f-  16b*,  arranging  ac 
cording  to  powers  of  a,  and  reducing. 

(5)  {5aib—4abc-\-6bc"—3a"cfz=z2oatb"-+16a-b2c"+36b-ci-{-9a4c'2 

—i0a3b"-c-[-60aib-ci—30a*bc 
—48ab°-c3+24a3bc~—36a-bc3. 
=25a4Z>3— 40asb-c+76a°-b-c~— 48a62c3 
4. 36b2c4— 30a4Z>c+  24a3Z>c3 
— 36a-bc3+9a*c°-. 
79.  Let  us  now  pass  on  to  the  extraction  of  the  squaro  root  of  algebraic 
quantities. 

Let  P  be  the  polynomial  whose  root  is  required,  and  let  R  represent  the 
root  which  for  the  moment  we  suppose  to  be  determined ;  let  us  also  suppose 
the  two  polynomials,  P  and  R,  to  be  arranged  according  to  the  powers  of 
6ome  one  of  the  letters  which  they  contain ;  a,  for  example. 

If  we  reflect  upon  the  law  just  given  of  the  formation  of  the  square  of  a 
polynomial,  it  will  be  seen  that  the  first  two  terms  of  the  polynomial  P,  when 
thus  arranged,  are  formed  without  reduction,  and  will  enable  us  at  once  to  de- 
termine the  first  two  terms  of  the  root  sought ;  for, 

1°.  The  square  of  the  first  term  of  R  must  involve  a,  affected  with  an  ex- 
ponent greater  than  any  that  is  to  be  found  in  the  other  terms  which  compose 
the  square  of  R  ;  because  this  exponent  is  double  the  highest  exponent  of  a  in 
R,  and  must  be  greater  than  the  doublo  of  any  lower  exponent,  or  than  the  re- 
sult produced  by  adding  it  to  one  of  the  lower  exponents,  or  by  adding  any 
two  of  them  together. 

2°.  Twice  the  product  of  tho  first  terrn  of  R  by  the  second  must  contain  a, 
affected  with  an  exponent  greater  than  any  to  be  found  in  the  succeeding 
terms ;  for  it  will  be  the  sum  of  tho  highest,  and  the  next  to  the  highest  ex 
ponent  of  a  in  R. 


76  ALGEBKA. 

It  follows  from  this,  that  if  P  bo  a  perfect  square, 

I.  Tho  first  term  roust  be  a  perfect  square  ;  and  the  square  root  of  this 
terra,  when  extracted  according  to  the  rule  for  monomials  I  Art.  49),  is  the  first 
term  of  R. 

II.  The  second  term  must  be  divisible  by  twice  the  first  term  of  R  thus 
found,  and  the  quotient  will  be  the  second  term  of  R. 

III.  In  order  to  obtain  the  remaining  terms  of  R,  square  the  two  terms  ofR 
already  determined,  and  subtract  the  result  from  P ;  we  thus  obtain  a  new 
polynomial,  P',  which  contains  twice  the  product  of  the  first  terra  of  R  by  the 
third  term,  together  with  a  series  of  other  terras.  But  twice  the  product  of 
the  first  term  of  R  by  the  third  must  contain  a,  affected  with  an  exponent 
greater  than  any  that  is  to  be  found  in  the  succeeding  terms,  and  hence  this 
double  product  must  form  the  first  term  of  P'.* 

IV.  The  first  term  of  P'  must  be  divisible  by  twice  the  first  term  of  R,  and 
the  quotient  will  bo  the  third  term  of  R. 

V.  In  order  to  obtain  the  remaining  terms  of  R,  square  tho  three  terms  of 
the  root  already  determined,  and  subtract  the  result  from  the  original  poly- 
nomial P;f  wo  thus  obtain  a  new  polynomial,  P",  concerning  which  we  may 
reason  precisely  in  tho  same  manner  as  for  P',  and  continuing  to  repeat  the 
operation  until  we  find  no  remainder,  we  shall  arrive  at  the  root  required. 

Tho  above  observations  may  be  collected  and  imbodied  in  the  following 

ROLE  FOR  THE  EXTRACTION  OF  THE  SQUARE  ROOT  OF  ALGEBRAIC  POLT 

NOMIALS. 

1°.  Arrange  the  polynomial  according  to  tiie  powers  of  some  one  letter. 

2°.  Extract  the  square  root  of  the  first  term  according  to  the  rule  for  monomi- 
al, and  the  result  will  he  the  first  term  of  the  root  required. 

3°.  Square  the  first  term  of  the  root  thus  determined,  and  subtract  it  from  the 
orig  inal  polynomial. 

4°.  Double  the  first  term  of  the  root,  and  divide  by  it  the  first  term  of  the  re- 
mainder, and  annex  the  result  (which  will  be  tiie  second  term  of  the  root),  with 
its  proper  sign,  to  the  divisor. 

5°.  Multiply  the  whole  of  this  divisor  by  tJie  second  term  of  the  root,  and  sub- 
tract the  product  from  the  first  remainder. 

G°.  Divide  this  second  remainder  by  twice  the  ram  of  the  first  two  terms  of 
the  root  already  found,  and  annex  the  result  (which  will  be  Oie  third  term  of 
tfie  root),  with  its  proper  sign,  to  the  divisor. 

7°.  Multiply  the  whole  of  this  divisor  by  the  Oiird  term  of  the  root,  and  sub- 
tract the  product  from  the  second  remainder  ;  continue  the  operation  in  this 
■manner  until  the  whole  root  is  ascertained. 

Tho  above  process  will  be  readily  understood  by  attending  to  the  following 
examples: 

r\  MirLE  1. 
Extract  tho  squaro  root  of  10r»— 10*3— l?r'+ ">r:4-P.   ---.V-f-1. 
Or,  arranging  according  to  tho  powers  of  r, 

*  Ti  of  the  second  term  of  11  usually  contains  tin-  tame  exponent  of  khe  Ictlur 

of  an  at,  but  this  is  already  subtracted  from  V,  and  not  left  Id  1'  . 

t  Ii.  .  this  operation  is  dispensed  with  by  following  the  p  In  the  fol- 
lowing rulo,  which  evidently  come  to  the  same  thing. 


SQUARE  ROOT  OF  POLYNOMIALS.  77 

Ox6  —  lSx6-^    10x*  —  10x3-f  5x2— 2x+l  3X3— 2x2+x— 1 

9X6 


6X3— 2x; 


— 12x5+    10x<  —lO-^+Sx2— 2x+l 
—12.^+      4x* 


6r»  —  4x2+  x 


6x*  —10xi-\-5x2—'2x-{-l 
Qx*  —  4r*+  a* 


63?  __  42;2+2z— 1 


—  Gx3-^2— 2x+l 

—  6x3+43:2— 2x+l 


0. 
Having  arranged  the  polynomial  according  to  powers  of  x,  we  first  extract 
the  square  root  of  Ox6,  the  first  term ;  this  gives  3x?  for  the  first  term  of  the 
root  required ;  this  we  place  on  the  right  hand  of  the  polynomial,  as  in  division ; 
squaring  this  quantity,  and  subtracting  it  from  the  whole  polynomial,  we  ob- 
tain for  a  first  remainder,  —  12xB+10x4 — 10x3-f-5x2 — 2x+l ;  we  now  double 
3a?,  and  place  it  as  a  divisor  on  the  left  of  this  remainder,  and  dividing  by  it 
— 12.1-5,  the  first  term  of  the  remainder,  we  obtain  the  quotient  — 2x2  (the 
second  term  of  the  root  sought),  which  we  annex,  with  its  proper  sign,  to  the 
double  root  6X3 ;  multiplying  the  whole  of  this  quantity,  6X3 — 2X2,  by  — 2xs 
(which  produces  twice  the  product  of  the  first  term  of  the  root  by  the  second, 
together  with  the  square  of  the  second),  and  subtracting  the  product  from  the 
first  remainder,  we  obtain  for  a  second  remainder,  Gx4 — lOx'+Sx2 — 2x+l. 
Next,  doubling  3x? — 2x2,  the  two  terms  of  the  root  thus  found,  and  dividing 
6x*,  the  first  term  of  the  new  remainder,  by  Gx3,  the  first  term  of  the  double 
root,  wo  obtain  x  for  a  quotient  (which  is  the  third  term  of  the  root  sought), 
and  annex  it  to  the  double  root  6X3 — 4x2,  multiplying  the  whole  of  this  quan- 
tity Gx3 — 4x2+x  by  x  (which  produces  twice  the  first  by  the  third,  twice  the 
second  by  the  third,  and  the  square  of  the  third),  and  subtracting  the  product 
from  the  second  remainder,  we  obtain  a  third  remainder,  — 6x3-r-4x2 — 2x+l ; 
we  now  double  3x3 — 2x2-f-:r>  the  three  terms  of  the  root  already  found,  and 
dividing  — Gx3,  the  first  term  of  the  new  remainder,  by  6X3,  the  first  term  of 
the  double  root,  we  obtain  — 1  for  the  quotient  (which  is  the  fourth  term  of 
the  root  sought),  and  annex  it  to  the  double  root  6X3 — 4x2-|-2x;  multiplying 
the  whole  of  this  quantity  6X3 — 4x2-f-2x — 1  by  — 1,  and  subtracting  it  from 
the  third  remainder,  we  find  0  for  a  new  remainder,  which  shows  that  the 
root  required  is 

3X3— 2x2+x— 1. 


78 


ALGEBRA. 


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SQUARE  ROOT  OF  POLYNOMIALS. 


79 


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80  ALGEBEA. 

80.  If  the  proposed  polynomial  contain  several  terms  affected  with  the  same 
power  of  tho  principal  letter,  we  must  arrange  the  polynomial  in  the  manner 
explained  in  division  (Art.  20) ;  and  in  applying  the  above  process  we  shall  be 
obliged  to  perform  several  partial  extractions  of  the  square  roots  of  tiie  coeffi- 
cients of  the  different  powers  of  the  principal  letter,  before  we  cau  arrive  at  the 
root  required. 

Extract  the  square  root  of 
(a-—2ab  +  bi)x*+2{a—b)(c—d)ri+\2{a  —  b){f-\-g)-\-(c—dy\3?+2{c—d) 

Ans.  (a—b)x-+(c—d)x+f+g. 

Such  examples,  however,  veiy  rarely  occur. 

Before  quitting  this  subject,  we  may  make  the  following  remarks : 

I.  No  binomial  can  be  a  perfect  square  ;  for  the  square  of  a  monomial  is  a 
monomial,  and  the  square  of  the  most  simple  polynomial,  that  is,  a  binomial, 
consists  of  three  distinct  terms,  which  do  not  admit  of  being  reduced  with 
each  other.  Thus,  such  an  expression  as  a--\-b2  is  not  a  square ;  it  wants  the 
term  ±2a6  to  render  it  the  square  of  (a±  b). 

II.  In  order  that  a  trinomial,  when  arranged  according  to  the  poire  s  of 
some  one  letter,  may  be  a  perfect  square,  the  two  extreme  terms  must  he  perfect 
squares,*  and  the  middle  term  must  be  equal  to  twice  the  product  of  the  square 
roots  of  the  extreme  terms.  When  these  conditions  are  fulfilled,  we  may  obtain 
the  square  root  of  a  trinomial  immediately,  by  the  following 

RULE. 

Extract  the  square  roots  of  the  extreme  terms,  and  connect  the  tico  terms  thus 
found  by  the  sign  -\-,  when  the  second  term  of  the  trinomial  is  positive,  and  by 
the  sign  — ,  when  the  second  term  of  the  trinomial  is  negative.  Thus,  the  ex- 
pression 

9a6— 48a*b2+64a?b* 
is  a  perfect  square ;  for  the  two  extreme  terms  are  perfect  squares,  and  the 
middle  term  is  twice  the  product  of  the  square  roots  of  the  extreme  terms; 
hence  the  square  root  of  the  trinomial  is 

i/thF—  y/~64a1b*. 
Or, 

3a?—8ab-. 

An  expression  such  as  4a2-\-12ab — db2  can  not  be  a  perfect  square,  although 
4a2  and  Ob",  considered  independently  of  their  signs,  are  perfect  squares,  and 
I2ab =2(2a  .  5b) ;  for  — Ob-  is  not  a  square,  sinco  no  quantity,  when  multi- 
plied by  itself,  can  have  the  sign  — 

III.  In  performing  the  operations  required  by  the  general  rule,  if  we  find 
that  tho  first  term  of  one  of  tho  remainders  is  not  exactly  divisible  by  twice 
the  first  term  of  the  root,  wo  may  immediately  conclude  that  the  polynomial 
is  not  a  perfect  square  ;  and  when  we  arrive  nt  a  term  in  the  root  having  a 
power  of  the  letter  of  arrangement  of  a  degree  less  than  half  that  of  this  letter 
in  (he  last  term  of  tho  giren  polynomial,  wo  ma\  be  sure  that  the  operation 
will  not  terminate.    This  is  on  tho  supposition  that  the  given  polynomial  is  ar- 

*  In  order  that  any  polynomial  maybe  n  perfect  square,  tin'  two  extremo  terms  must  be 
perfect  squares,  If  it  be  nrr:>  oordlng  to  the  powi  ra  of  some  letter. 


CUBE  HOOT  OF  POLYNOMIALS.  81 

ranged  according  to  the  decreasing  powers  of  the  letter.  If  it  be  according  to 
the  increasing  powers,  substitute  the  word  greater  for  "  less"  in  the  above 
precept. 

IV.  We  may  apply  to  the  square  roots  of  polynomials  which  are  not  per- 
fect squares  the  simplifications  already  employed  in  the  case  of  monomials 
(Art.  51).     Thus,  in  the  expression 

VV&  +  4a263-r.4a&3. 

The  quantity  under  the  radical  sign  is  not  a  perfect  square,  but  it  may  be 
put  under  the  form 

^/ab{a--{-4ab-r-Ab'~). 

The  factor  within  brackets  is  manifestly  the  square  of  a -\-2b;  hence 

Vra3&-f4a263+4a63=  Vab{a--irAab-ir4b-) 

=  Vabla+^f 
=  (a+2b)  j~ab. 

81.  Let  us  next  proceed  to  form  the  cube  of  x-\-a. 

(x-f  a)3=(.r+a)  X  (x+a)  X  (x+a) 

=x3-{-3x'2a-{-3xa2-{-a3  by  rules  of  multiplication. 

Let  it  be  required  to  form  the  cube  of  a  trinomial  (x-\-a-\-b)',  represent 
the  last  two  terms  a-\-b  by  the  single  letter  s ;  then 

{x+a+b)*=:(x  4-s)3 

=x*+3x'2  s-f-3xs24-s3 
=xs+3x°{a+ Z,)-f-3.r(a+Z>)2+(a  +  6)3 
=.r3+3.r2  a-f3x26-f  3.ra2+6.ra&  +  3:r&2-fa3 
-j-3^2  b  +  3ab*+ V. 
This  expression  is  composed  of  the  sum  of  the  cubes  of  all  Hie  terms,  together 
with  three  times  the  sum  of  the  squares  of  each  term,  multiplied  by  the  simple 
power  of  each  of  the  others  in  succession,  together  with  six  times  the  product  of 
the  simple  power  of  all  Hie  terms- 

By  following  a  process  of  reasoning  analogous  to  that  employed  in  (Art.  78), 
we  can  prove  that  the  abovo  law  of  formation  will  hold  good  for  any  polynomial 
of  whatever  number  of  terms.     We  shall  thus  find 
(a+6+c+d)3     =a3  +  i34-c3+rf3+3a26  +  3a2c+3a2J+362a+3t-c+362rf 

+  3c*a+3c-b  +  3cH+3dia-\-3d%+3d~c-\-Gabc+Gabd-\-6acd+Gbcd 
(2a'—4ab-\-3b''Y=  8a6  —  64a353  +  27b6  —  48a5i  +  36a4b-  -4-  96a4/r  +  U-ia-b* 

_|_54„:/,i_108a/;5— 144a363 
—8a6—48a5b-lr132a*bi—208asb3-\-198a2b-i  —  108ab5-\-27b\ 

In  a  similar  manner,  we  can  obtain  the  4th,  5th,  &c,  powers  of  any  poly- 
nomial. 

For  more  upon  this  subject,  see  a  subsequent  article  (105). 

82.  We  shall  now  explain  the  process  by  which  we  can  extract  the  cube 
root  of  any  polynomial,  a  method  analogous  to  that  employed  for  the  square 
root,  and  which  may  easily  be  generalized,  so  as  to  be  applicable  to  the  ex- 
traction of  roots  of  any  degree. 

Let  P  be  the  given  polynomial,  R  its  cube  root.  Let  these  two  poly- 
nomials be  arranged  according  to  the  powers  of  some  one  letter,  a,  for  example. 
It  follows,  from  the  law  of  formation  of  the  cube  of  a  polynomial,  that  the  cubo 
of  R  contains  two  terms,  which  are  not  susceptible  of  reduction  with  any 
others ;  these  are,  the  cube  of  the  first  term,  and  three  times  the  square  of 

F 


12  ALGEBRA 

tho  first  term  multiplied  by  the  second  term;  for  il  is  manifest  that  these  fw  D 
terms  will  involve  a  affected  with  an  exponent  higher  than  any  that  is  to  oe 
found  in  the  succeeding  terms.  Consequently,  these  two  terms  must  form 
the  first  two  terms  of  P.  Hence,  if  we  extract  the  cube  root  of  the  first  term 
of  P,  we  shall  obtain  the  first  term  of  R,  and  then,  dividing  the  second  term 
of  P  by  three  times  tho  square  of  the  first  term  of  R  thus  found,  the  quotient 
will  be  the  second  term  of  R.  Having  thus  determined  the  first  two  terms  of 
R,  cube  this  binomial,  and  subtract  it  from  P.  The  remainder,  P',  being  ar- 
ranged, its  first  term  will  be  three  times  the  product  of  the  square  of  the  first 
term  of  R  by  tho  third,  together  with  a  series  of  terms  involving  a,  affected 
with  a  less  exponent  than  that  with  which  it  is  affected  in  this  product. 
Dividing  tho  first  term  of  P'  by  three  times  the  square  of  the  first  term  of  R, 
the  quotient  will  be  the  third  term  of  R.  Forming  the  cube  of  the  trinomial 
root  thus  determined,  and  subtracting  this  cube  from  the  original  polynomial 
P,  we  obtain  a  new  polynomial,  P",  which  we  may  treat  in  the  same  manner 
as  P',  and  continue  the^operation  till  the  whole  root  is  determined.* 

EXAMPLES. 

(1)  Extract  the  cube  root  of  27Z3— 135a:2+225x— 125. 

(2)  V(8^+48zx5+60;2r4— mz*J?— 90r4.c-+108c5x— 27z6). 

ANSWERS. 

(1)  3r— 5.  |  (2)  2x2+4;.r— 3z3. 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  NUMBERS. 
83.  Rules  are  given  in  Arithmetic  for  extracting  Ac  square  and  cube  roots  ol 
any  proposed  number;  we  shall  now  proceed  to  explain  the  principles  upon 
which  these  rules  are  founded. 
The  numbers 

1,  2,  3,   4,    5,    6,    7,    8,    9,    10,    100,       1000, 

when  squared,  become 

1,  4,  9,  1G,  25,  3G,  49,  64,  81,  100,  10000,  1000000, 

and  reciprocally,  the  numbers  in  the  first  line  are  the  square  roots  of  the  nura 
bers  in  the  second. 

Upon  inspecting  these  two  lines  we  perceive  that,  among  numbers  expressed 
by  one  or  two  figures,  there  are  only  nine  which  arc  the  squares  of  other 
whole  numbers;  consequently,  the  square  root  of  all  otht>r  numbers  consisting 
of  one  or  two  figures  must  be  a  whole  Dumber  pins  a  fraction. 

Thus,  the  square  root  of  53,  which  lies  between  19  and  64,  is  7  plus  a  frac- 
tion.    So,  also,  tho  square  root  of  91  is  9  plus  a  fraction. 

84.  It  is,  however,  very  remarkable  Omt  the  square  root  of  a  whole  number, 
which  is  not  a  perfect  square,  can  not  be  ,  rt>r,  m  d  art  fraction,  and  is, 

therefore,  incommensurable  with  unity. 

To  prove  this,  lot  -r,  a  fraction  in  its  lowest  terms,  1"-.  if  possible,  the  square 

a  ./-' 

root  of  some  whole  Dumber;  thou  the  square  of  j,  or    .  must  b.«  equal  to  tins 

whole  iiunih.-r.     But  since  a  and  l>  are,  by  supposition,  prime  to  each  other 

•  Tliis  Bttbject  wi'l  be  resumed  a  f.'u  pa  ei  farther  an. 


SQ.UARE  ROOT  OF  NUMBERS.  83 

{i.  e.,  have  no  common  divisor),  a2  and  b2  are  also  prime  to  each  other;*  there- 
at 

fore  7-  is  an  irreducible  fraction,  and  can  not  be  equal  to  a  whole  number. 
b* 

85.  The  difference  between  the  squares  of  two  consecutive  whole  numbers 
is  greater  in  proportion  as  the  numbers  themselves  are  greater ;  the  expres 
sion  for  this  difference  can  easily  be  found. 

Let  a  and  a-|-l  be  two  consecutive  whole  numbers ; 
Then, 

(a+1)2         =a3+2a+l. 
Hence, 

(a+1)2— a3=2a+l; 

that  is  to  say,  the  difference  of  the  squares  of  two  consecutive  whole  numbers  u 
equal  to  twice  the  less  of  the  two  numbers  plus  unity. 

Thus,  the  difference  between  the   squares  of  348  and  347  is   equal  to 

2x347  +  1,  or  695. 

• 

*  This  depends  upon  the  principle  that,  if  any  prime  number,  P,  will  divide  the  product 

of  two  numbers,  it  must  divide  one  of  them,  which  may  be  demonstrated  as  follows  : 
Let  A  and  B  be  the  two  numbers,  and  let  it  be  sujTposed  that  P  will  not  divide  A,  we 

are  to  prove  that  it  must  divide  B. 

Dividing  A  by  P,  and  denoting  the  quotient  by  Q,  and  the  remainder  by  P',  we  have 

AB  P'B 

A=PQ+P'  .-.multiplying  by  B,  AB=PQ.B+P'B  ,\  dividing  by  P,  —  =  Q.B+— 

Since  by  hypothesis  AB  is  divisible  by  P,  P'B  must  be,  else  we  should  have  a  whole 
number,  equal  to  a  whole  number  plus  a  fraction,  which  is  impossible.  Proceed  now  with 
P  and  P'  after  the  method  for  finding  a  common  divisor,  and  let  P",  P'",  &c,  be  the  suc- 
cessive remainders,  which  can  none  of  them  be  zero,  because  P  is  by  hypothesis  a  prime 
number  (i.  e.,  a  number  divisible  only  by  itself  and  unity) :  these  remainders  must  go  on  di 
minishing  till  the  last  becomes  ujiity,  and  we  shall  have  the  series  of  equalities, 

P=P'a'+P".  P'=P" Q/'+P'",  &C. ; 
or,  multiplying  by  B  and  dividing  by  P, 

_     P'Q'B  ,  P"B  P'B     P"Q/'B  ,  P'"B 

B=-p-+-p--  -r=~p — i"p-« &c- 

The  first  of  these  equalities  shows  that  if  P'B  is  divisible  by  P,  P"B  must  also  be  divisi- 
ble ;  and  if  both  these  are  divisible,  the  second  equality  shows  that  P'"B  is  divisible  by 
P,  and  so  on.  But  the  remainders,  P",  P"',  &c,  diminish  till  the  last  becomes  unity,  and 
we  shall  thus  have,  finally,  1XB,  or  B  divisible  by  P.  Q_.  E.  D. 

Now,  since  a2  is  the  product  of  a  and  a,  any  prime  number  which  divides  a2  must  divide 
a,  or  which  divides  b2  must  divide  b,  so  that  any  prime  number  which  divides  both  a-  and 
b2  must  divide  a  and  b. 

Every  number  is  either  prime  or  composed  of  prime  numbers  as  factors,  and  if  this  nura 
ber  will  divide  the  two  terms  of  a  fraction,  its  prime  factors  will  successively  divide  them 
This  follows  from  (10,  I.,  2). 

As  an  addition  to  this  note  may  be  demonstrated  the  following  theorem :  A  literal  quart 
lily  can  not  be  decomposed  into  prime  factors  in  different  ways. 

Let  ABCD...  be  a  product  of  prime  factors,  and  suppose  that  it  could  be  equal  to  an 
other  product,  abed . . .,  the  factors  a,  b,  c,  d. . .  being  also  prime.  The  factor  a,  dividing 
abed,  must  divide  the  equal  ABCD . . . ;  but  if  the  prime  quantity  a  is  different  from  each 
of  the  quantities  A,  B,  C,  D,  &c,  it  can  not  divide  any  of  them.  Not  dividing  either  A  or 
B  according  to  the  above  theorem,  it  can  not  divide  the  product  AB.  Not  dividing  either 
AB  or  C,  it  will  not  divide  the  product  ABC.  and  so  on.  The  factor  a  must,  therefore, 
necessarily  be  equal  to  one  of  the  factors  A,  B,  C,  <Scc.  Suppose  «=A.  Dividing  the  two 
products  by  A,  the  remaining  products,  BCD  . . .  and  bed  .  ■ .,  are  still  equal,  and  applying  to 
them  the  preceding  reasoning,  we  conclude  that  b  ought  to  be  equal  to  one  of  the  factors  of 
the  product,  BCD...,  and  so  on.  The  two  products,  ABCD...  and  abed...,  must,  there- 
fore, be  composed  of  the  same  prime  factors.  Q_  E.  D 


7fc 


34  ALGEBRA- 

Tlio  square  of  a  number  will  always  consist  of  twice  as  many  digits,  or  om 
iess  than' twice  as  many,  as  the  number  itself.  Thus,  the  square  of  10  is  100, 
and  the  square  of  any  number  less  than  10  must  be  less  than  100,  or  contain 
not  more  than  two  6gures.  The  square  of  100  is  10000,  and  the  square  of  all 
numbers  between  10  and  100  must  be  between  100  and  10000;  i.  c,  consist 
of  3  or  4  figures.  In  the  same  way  it  may  be  shown  that  the  square  of  a 
number  containing  three  figures  must  be  one  containing  five  or  six  figures,  and 
so  on;  i.  < ..  the  square  of  a  number  consists  of  twice  as  many  digits  as  the 
number  itself,  or  one  less  than  twice  as  many. 

Let  us  now  proceed  to  investigate  a  process  for  the  extraction  of  the  square 
root  of  any  Dumber,  beginning  with  whole  numbers. 

EXTRACTION  OF  THE  SdUARE  ROOT  OF  WHOLE  NUMBERS 

86.  If  the  number  proposed  consist  of  one  or  two  figures  only,  its  root  ma> 
oe  found  immediately  by  inspecting  the  squares  of  the  nine  first  numbers  in 
Art.  83).  Thus,  the  square  root  of  25  is  5,  the  square  root  of  42  is  6  plus  a 
fraction,  or  G  is  the  approximate  square  root  of  42,  and  is  within  one  unit  of 
the  true  value ;  for  42  lies  between  36,  which  is  the  square  of  G,  and  49,  which 
is  the  square  of  7. 

Let  us  consider,  then,  a  number  composed  of  more  than  two  figures,  6084 
for  example. 

Since  this  number  consist  of  four  figures,  its  root  must  60'84 

necessarily  consist  of  two  figures,  that  is  to  say,  of  tens  49 

and  units.     Designating  the  tens  in  the  root  sought  by  a,         148 
and  the  units  by  b,  we  have 

6084  =  (a+t)-=a2-r-2a&4-&2, 
which  shows  that  the  square  of  a  number  consisting  of  tens  and  units  is  com- 
posed  of  the  square  of  the  tens,  plus  twice  the  product  of  the  tens  by  the  ui 
phis  the  square  of  the  units. 

This  being  premised-,  since  the  square  of  a  certain  number  of  teus  must  be 
a  certain  number  of  hundreds,  or  have  two  ciphers  on  the  right,  it  follows  that 
the  squares  of  tho  tens  contained  in  the  root  must  be  found  in  the  part  60  (or 
60  hundreds),  to  the  left  of  the  last  two  figures  of  6081  (which  written  at  full 
length  is  6000 -(-804-4),  tho  81  forming  do  part  of  the  square  of  the  tens;  we, 
therefore,  separate  the  last  two  figures  from  the  others  by  a  point.  The  part 
60  is  comprised  between  the  two  perfect  squares  19,  and  64f  the  roots  of  which 
are  7  and  8;  hence  7  is  tho  figure  which  expresses  the  number  of  tens  in  the 
root  soufiht :  for  6000  is  evidently  comprised  between  1900  and  6400,  which 
are  the  squares  of  70  and  80,  and  tho  root  of  6084  must,  therefore,  be  com- 
prised between  7(1  and  80;  hence,  the  rOOl  BOUghl  18  Composed  of  7  tens  and 
u  certain  number  of  units  less  than  ten. 

The  figure  7  being  'bus  found,  we  place  it  on  the  righl  of  the  given  Dumber, 
in  the  place  often*,  separated  by  e  vertical  line  as  in  division;  we  then  sub- 
tract   in,  whieb    is   the   square   of  7.  from   60,  which    leaves  as   remainder  11 

(which  is  11  hundreds),  after  which  we  write  the  remaining  figures,  B4« 
Having  taken  away  the  Bquare  of  the  tens,  the  remainder,  1184,  contains,  as 
w»«  have  Been  above,  twice  the  product  of  the  tens  multiplied  by  the  units 
plus  the  square  of  the  units.     But  the  product  of  the  tens  multiplied  by  the 

units    must  be   tens,  CT  have    one  cipher  on  the  right,  iind,  therefore,  the  last 


L18'4 

118'4 


0. 


SQUARE  ROOT  OP  NUMBERS.  85 

figure  4  can  not  form  any  part  of  the  product  of  the  tens  by  the  units;  we, 
therefore,  separate  it  from  the  others  by  a  point. 

1 1' we  double  the  tens,  which  gives  14,  and  divide  the  118  tens  by  14,  the 
quotient  8  is  the  figure  of  units  in  the  root  sought,  or  a  figure  greater  than  the 
one  required.  It  may  manifestly  be  greater  than  the  figure  sought,  for  118 
may  contain,  in  addition  to  twice  the  product  of  the  tens  by  the  units,  other 
tens  arising  from  the  square  of  the  units,  which  may  exceed  the  denomination 
units.  In  order  to  determine  Whether  8  expresses  the  real  number  of  units 
in  the  root,  it  is  sufficient  to  place  it  on  the  right  of  14,  and  then  multiply  the 
number  148,  thus  obtained,  by  8.  In  this  manner  we  form,  1",  the  square  of 
the  units  ;  2°,  twice  the  product  of  the  units  by  the  tens.  This  operation 
being  effected,  the  product  is  1184;  subtracting  this  product,  the  remainder  is 
0,  which  shows  that  6084  is  a  perfect  square,  and  78  the  root  sought. 

It  will  be  seen,  in  reviewing  the  above  process,  that  we  have  successively 
subtracted  from  G084,  the  square  of  7  tens  or  70,  plus  twice  the  product  of  70 
by  8,  plus  the  square  of  8,  that  is,  the  three  parts  which  enter  into  the  com- 
position of  the  square  of  70-f-8,  or  78  ;  and  since  the  result  of  this  subtraction 
is  0,  it  follows  that  G084  is  the  square  of  78. 

The  quotient  obtained  from  dividing  by  double  the  tens  is  a  trial  figure  ;  it 
will  never  bo  too  small,  but  maybe  too  great,  and  on  trial  may  require  to  be  di- 
minished by  one  or  two  units. 

Take  as  a  second  example  the  number  841.  8'41  29 

This  number  being  comprised  between  100  and  10000,  its 


44'1 
441 


0. 


root  must  consist  of  two  figures,  that  is  to  say,  of  tens  and  49 
units.  We  can  prove,  as  in  the  last  example,  that  the  root 
af  the  greatest  square  contained  in  8,  or  in  that  portion  of  the 
number  to  the  left  of  the  last  two  figures,  expresses  the  number  of  tens  in  the 
root  required.  But  the  greatest  square  contained  in  8  is  4,  whose  root  is  2, 
which  is,  therefore,  the  figure  of  the  tens.  Squaring  2,  and  subtracting  the 
result  from  8,  the  remainder  is  4 ;  bringing  down  the  figures  of  the  second 
period  41,  and  annexing  them  on  the  right  of  4,  the  result  is  441,  a  number 
which  contains  twice  the  product  of  the  tens  by  the  units,  plus  the  square  of 
the  units. 

We  may  farther  prove,  as  in  the  last  case,  that  if  we  point  off  the  last  figure 
I,  and  divide  the  preceding  figures  44  by  twice'the  tens,  or  4,  the  quotient 
will  be  either  the  figure  which  expresses  the  number  of  units  in  the  root,  or  a 
figure  greater  than  the  one  sought.  In  this  case  the  quotient  is  11,  but  it  is 
manifest  that  we  can  not  have  a  number  greater  than  9  for  the  units,  for  other- 
wise we  must  suppose  that  the  figure  already  found  for  the  tens  is  incorrect. 
Let  us  tiy  9 ;  place  9  to  the  right  of  4,  and  then  multiply  this  number  49  by 
9 ;  the  product  is  441,  which,  when  subtracted  from  the  result  of  the  first 
operation,  leaves  a  remainder  0,  proving  that  29  is  the  root  required. 

Let  us  take,  as  a  third  example,  a  number  which  is  not  a  perfect  square, 
such  as  1287. 

Applying  to  this  number  the  process  described  in  the  pre-  12'87  35 

ceding  example,  we  find  that  the  root  is  35,  with  a  remainder  9 

62.     This  shows  that  1287  is  not  a  perfect  square,  but  that        65 
it  is  comprised  between  the  square  of  35  and  that  of  36. 
rims,  when  the  number  is  not  a  perfect  squai  e,  the  above 


38'7 
325 


62 


56'£2'14'44 
49 

145 

78'2 
725 

1503 

571 '4 
4509 

15068  12054'4 
120544 

0. 

86  ALGEBRA. 

process  enables  us  at  least  to  determine  the  root  of  llie  greatest  square  job 
tained  in  the  number,  or  the  integral  part  of  the  root  of  the  number. 

87.  Let  us  pass  on  to  consider  the  extraction  of  the  square  root  of  a  num 
ber  composed  of  more  than  four  figures. 

Let  56821444  be  the  number.  56'ir2'14'44  7538 

Since  the  number  is  greater  than  10000,  its  root 
must  be  greater  than  100  ;  that  is  to  say,  it  must 
consist  of  more  than  two  figures.*  But,  whatever 
the  number  may  be,  we  may  always  consider  it  as 
composed  of  units  and  of  tens,  the  tens  being  ex- 
pressed by  one  or  more  figures.  (Thus,  any  num- 
ber such  as  37142  may  be  resolved  into  37140  +  2, 
or  3714  tens,  plus  two  units.) 

Now  the  square  of  the  root  sought,  that  is,  the  proposed  number,  contains 
the  square  of  the  tens,  plus  twice  the  product  of  the  tens  by  the  units,  plus 
the  square  of  the  units.  But  the  square  of  the  tens  must  give  at  least  hun- 
dreds ;  hence  the  last  two  figures,  44,  can  form  no  part  of  it,  and  it  is  in  the 
portion  of  the  number  to  the  left  hand  that  we  must  look  for  that  square. 
But  this  portion  containing  more  than  two  figures,  its  root  will  consist  of  units 
and  tens  ;  it  will,  therefore,  be  necessary  to  commence  the  process  for  finding 
the  root  of  this  portion  by  cutting  off  its  two  right-hand  figures,  14,  and  the 
square  of  the  tens  of  the  tens  is  to  be  sought  in  tho  figures  now  remaining  at 
the  left,  5682.  This  number  being  the  square  of  two  figures,  we  again  separate 
82,  and  seek  for  the  square  of  the  tens  of  the  tens  of  the  tens  in  the  two  re- 
maining figures,  56.  The  given  number  is  thus  separated  into  periods  of  two 
figures  each,  beginning  on  the  right.  We  then  go  on  to  extract  the  root  of 
the  number  5682,  as  in  the  previous  examples;  this  will  give  the  tens  of  the  root 
of  the  number  568214.  We  then  double  these  tens  for  a  divisor,  and  take  the 
remainder  after  the  last  operation,  with  14  annexed  for  a  dividend  ;  we  divide 
this  dividend,  after  cutting  off  the  right-hand  figure,  and  the  quotient  will  be 
the  units  of  the  root  of  568214.  All  the  figures  now  found  of  the  root  will 
constitute  the  tens  of  the  root  of  the  given  number,  and  we  find  the  units  by 
the  rule  previously  given.     The  detail  of  the  whole  operation  is  as  follows  : 

Extracting  tho  root  of  56,  we  find  7  for  the  root  of  49,  the  greatest  square 
contained  in  56;  we  place  7  on  the  righl  of  the  proposed  Dumber,  and  squaring 
it,  subtract  49  from  56,  which  gives  a  remainder  7,  to  which  we  annex  the  fol- 
lowing period,  82.  Separating  the  last  figure  to  tin-  right  of  782,  and  then 
dividing  78  by  14,  which  is  twice  the  root  already  found,  we  have  5  for  a  quotient, 
which  we  annex  to  14;  we  then  multiply  the  whole  Dumber  115  by  5,  and 
subtract  tho  product  725  from  782.  We  next  bring  down  the  period  11,  an- 
nex it  to  the  second  remainder  57,  ami  point  off  the  last  BgUTO  of  this  number 
5714.  Dividing  571  by  150,  which  is  twice  tho  root  already  found,  the  quotient 
is  3,  which  we  place  to  the  right  of  150,  and  multiplying  the  whole  nuiiher 
1503  by  3,  we  subtract  the  product  4509  from  .r>?  1  I. 

Finally,  we  bring  down  the  last  period  1 1,  annex  it  to  the  third  remaindei 
1205,  and  point  elf  the  last  figure  of  this  Dumber  12054  I.     Dividing  L2054  Uy 

*  Wo  liPfvo  Been  in  tin:  lust  article  that  it  will  lialf  as  many  as  tl)« 

given  comber.    Bad  die  given  number  contained  bat  i  an  i,  the  root  would  still  he 

composed  of  four 


SQUARE  ROOT  BY  APPROXIMATION.  87 

150G,  which  is  twice  the  root  already  found,  the  quotient  is  8,  which  we  place 
yn  the  right  of  1506,  and  multiplying  tho  whole  number  15068  by  8,  we  sub- 
tract the  product  120544  from  the  last  result  120544.  The  remainder  is  0  ; 
hence  7538  is  the  root  sought. 

From  what  has  been  said  above,  it  is  easy  to  deduce  the  rule,  ordinarily 
given  in  Arithmetic,  for  the  extraction  of  the  square  root  of  a  number  consist- 
ing of  any  number  of  figures,  and  which  it  is  unnecessary  here  to  repeat. 

EXTRACTION  OF  THE  SQUARE  ROOT  BY  APPROXIMATION. 

88.  When  a  whole  number  is  not  tho  square  of  another  whole  number,  we 
have  seen  (Art.  84)  that  its  root  can  not  be  expressed  by  a  whole  number  and 
an  exact  fraction  ;  but  although  it  is  impossible  to  determine  the  precise  value 
of  the  fraction  which  completes  the  root  sought,  we  can  approximate  it  as 
nearly  as  we  please. 

Suppose  that  a  is  a  whole  number  which  is  not  a  perfect  square,  and  that 

we  are  required  to  extract  the  root  to  within  — ,  that  is,  to  determine  a  number 

which  shall  differ  from  the  true  root  of  a,  by  a  quantity  less  than  the  fraction  — . 

To  effect  this,  let  us  observe  that  the  quantity  a  may  be  put  under  the  form 

an* 

—5- ;  if  we  designate  the  integral,  or  whole  number,  portion  of  the  root  of  an1 

an1 
by  r,  this  number  an?  will  be  comprised  between  r2  and  (r+1)- ;  hence,  — - 

r"-         (r+1)2 
is  comprised  between  —  and — ,  and  consequently,  the  root  of  a  is  com- 

r2         (r+1)2  r         r+1 

prised  between  the  roots  of  —  and : — ,  that  is,  between  -  and .     Thus, 

r  n2  n2  n  n 

r  1 

it  appears  that  —  represents  the  square  root  of  a  within  -  of  the  true  value 

From  this  we  derive  the  following 

RULE. 

To  extract  the  square  root  of  a  whole  number  to  within  a  given  fraction,  mul- 
tiply the  given  number  by  the  square  of  the  denominator  of  the  given  fraction  ; 
extract  the  integral  part  of  the  square  root  of  the  product,  and  divide  this  in- 
tegral part  by  the  given  denominator. 

Let  it  be  required,  for  example,  to  find  the  square  root  of  59  within  TV  of 
the  true  value. 

Multiply  59  by  the  square  of  12,  that  is,  144,  the  product  is  8496  ;  the  in- 
tegral part  of  the  root  of  8496  is  92.  Hence  f  §  or  7^  is  the  approximate  root 
of  59,  the  result  differing  from  the  true  value  by  a  quantity  less  than  -i. 

So,  also, 

Vll   =  3T4j  true  to-Jj, 
-/223=142J  true  to  ^. 

89.  The  method  of  approximation  in  decimals,  which  is  the  process  most 
frequently  employed,  is  an  immediate  consequence  of  the  preceding  rule. 

In  order  to  obtain  the  square  root  of  a  whole  number  within  J^,  T  L,  jJ^  .  .  . 
of  the  true  value,  we  must,  according  to'  the  above  rule,  multiply  the  proposed 
number  by  (10)2,  (100)2,  (1000)2, or,  which  comes  to  the  same  thing, 


88  ALGEBRA. 

place  to  the  right  of  the  number,  two,  four,  six, ciphers,  then  extract 

the  integral  part  of  the  root  of  the  product,  nod  divide  the  result  by  10,  100, 
1000 

Hence,  in  order  to  obtain  any  required  number  of  decimals  in  the  root,  we 
must 

Place  on  the  right  hand  of  Oie  proposed  number  twice  as  many  zeros  as  we 
loish  to  have  decimal  figures ;  extract  the  integral  part  of  the  root  of  this  new 
number,  and  then  marlc  off  in  the  result  the  required  number  of  decimal  places 

EXAMPLES. 

(1)  Extract  the  square  root  of  3  to  six  places  of  decimals. 

Ans.  1.732050. 

(2)  Extract  the  square  root  of  5  to  six  places  of  decimals. 

Ans.  2.236068. 

(3)  Extract  the  square  root  of  12  to  six  places  of  decimals. 

Ans.  3.464101. 
When  half,  or  one  more  than  half,  the  figures  are  found,  the  rest  may  be 
found  by  division. 

(4)  Extract  the  square  root  of  2  to  nine  places  of  decimals. 

The  first  five  figures  of  the  root  found  by  the  ordinary  method  are  1.41 1J  . 
with  the  remainder,  3836.  The  next  divisor  is  28284.  Dividing  3836  by 
28284,  according  to  the  ordinary  method  of  division,  produces  1356  for  a  quo- 
tient, which,  annexed  to  1.4142,  befoi-e  found,  gives  for  the  root  required 
1.41421356.* 

Extract  the  square  root  of  11  to  six  places  of  decimals. 

Ans.  3.316624. 

EXTRACTION  OF  THE  SQ.UARE  ROOT  OF  FRACTIONS. 

la      V« 

We  have  seen  (Art.  62)  that  -   T=~ 7r»  hence,  in  order  to  extract  tne 

V  o       y  0 

square  root  of  a  fraction,  it  is  sufficient  to  extract  the  square  roots  of  the  numer- 
ator and  denominator,  and  then  divide  the  former  result  by  the  latter.  This 
method  may  be  employed  with  advantage  when  either  one  or  both  of  the  terms 
of  the  proposed  fraction  aro.  perfect  squares;  but  when  this  ia  not  the  case,  it 
will  bo  found  inconvenient  in  practice.     If,  for  example,  we  take  the  fraction 

/3      V~3 
|,  although  .. /-=~=  (since  each  ot  these  expressions,  when  multiplied  by  it- 
self, produces  the  same  quantity.  '•;),  we  must  find  an  approximate  value  both 

for  <J?>  anu>  a's0  fur  V&i  il,,,l'  il'u'1'  il"'  Av<'  shall  not  be  able  to  determine  at 

once  tho  degree  of  approximation  in  the  result.     Under  such  circumstances 

the  following  process  may  bo  emplo]  ed  : 

a  al> 

Let  tho  proposed  fraction  bo  j,  this  may  be  put   under  the  form  y-;  this 

feeing  premised,  let  r  represent  the  integral  part  of  the  root  of  the  numeratos 

*  The  reason  tor  this  rulo  may  be  •■\\>-u  thus :  Let  k  be  the  part  <>f  the  root  already 
found,  and  z  the  remaining  part  Then  /.--(--  will  be  the  whole  root,  and  (A--f-r)-=A-'-f-2£; 
-j-:-  the  given  number;  aa  i  if  hut  a  small  fraction  of  k,  :■  w  ill  be  a  still  smaller  fraction, 
ami  may  be  di  looted,  so  that  the  given  camber  nay,  without  sensible  error,  he  considered 
eqaal  to  /.--f--/.r.  Bat  k*  baa  bean  taken1  away,  and  the  remainder,  ••'A:,  divided  i>\ 
i  .-. 


SQ.UARE  HOOT  OF  FKACTIO:  89 

ab;  hence  tj,  <tr  y,  is  comprised  between  -j-  and  — .-, — ;  consequently,  the 

d  y  T-l—  1  T 

root  of  j  is  comprised  between  t  and  — ?— .     Thus,  it  appears  that  r  repre- 

a  1 

Bents  the  root  of  -r  within  7-  of  the  true  value.     Hence,  in  order  to  obtain  the 
0  b 

square  root  of  a  fraction, 

Make  the  denominator  of  the  fraction  a  perfect  square,  by  multiplying  botli 
terms  of  the  fraction  by  the  denominator ;  extract  the  integral  part  of  the  root  of 
Uie  numerator,  and  divide  the  result  by  the  denominator. 

Let  it  be  required  to  extract  the  square  root  of  T7j. 

7  X  13  91 

This  fraction  is  the  same  as  ,  or  .     But  the  integral  part  of  the 

9 
square  root  of  91  is  9 ;  hence  —  is  the  root  so,    lit,  a  result  within  ^  of  the 

true  value. 

A  greater  degree  of  approximation  may,  perhaps,  bo  required.     In  this  case, 

91 
returning  to  the  number ,  extract  the  root  of  91  to  any  required  degree 

of  approximation.     Suppose,  for  example,  we  wish  to  find  the  root  of  91  within 

—  of  the  real  value,  it  will  become  by  (Ait.  88)  -\/91=9  .  53  . . . .     Hence 

7  91  9.53  1 

the  root  of — ,  or ,  will  be  — - — ,  equal  -73  within of  the  true  value. 

13        (13)*  13  1300 

Remark. — It  frequently  happens  that  the  denominator  of  the  fraction,  al- 
though not  a  perfect  square,  has  a  perfect  square  for  one  of  its  factors,  in 
which  case  the  above  operation  may  be  simplified. 

23 
Let  the  fraction,  for  example,  be  — .     48  is  equal  to  16x3,  or  (4)2x3; 

23  X  3 
hence,  multiplying  both  terms  of  the  fraction  by  3,  it  becomes  7- - — j—,  or 

69 
tt— — ;  and  the  denominator  is  thus  made  a  perfect  square.     Extracting  the 

1  R   3  83 

root  of  69  to  — ,  which  eives  8 . 3,  we  find  — '— ,  or for  the  root  required,  a 

10  s  12  120  H 

result  within  — ■  of  the  true  value. 

In  general,  therefore,  whenever  the  denominator  of  the  fraction  involves  a 

factor  which  is  a  perfect  square,  multiply  both  terms  of  the  fraction  by  the  factor 

which  is  not  a  perfect  square. 

_  5  1 

.hxrract  the  square  root  of-  to  within  — . 
1  6  48 

5=5x6x8"=i920_     — =43  «« 

6        6s  X  8J        6Z  X  82  V  6     48 

EXTRACTION  OF  THE  SOUARE  ROOT  OF  DECIMAL  FRACTIONS. 

90.  This  process  is  an  immediate  consequence  of  the  preceding  remark. 
Required,  for  example,  the  square  root  of  2 .  36. 


90  ALGEBRA. 

236 
This  fraction  is  the  same  as  y— ;  in  this  case  the  denominator  is  a  perfect 

square  ;  extracting,  therefore,  the  integral  part  of  the  root  of  the  numerator,  we 

15  1 

have  — ,  a  result  within  —  of  the  time  value. 
10  10 

Again,  let  it  be  required  to  extract  the  square  root  of  3.425. 

3425 

This  fraction  is  the  same  as  .     But  1000  is  not  a  perfect  square  ;  it  is, 

however,  equal  to  100x10,  or  (10):Xl0;  thus,  in  order  to  render  the  de- 
nominator a  perfect  square,  it  is  sufficient  to  multiply  both  terms  of  the  frac- 

34250         34250      _ 
tion  by  10,  which  gives  ,  or  ,        ...     Extracting  the  integral  part  of  the 

185 
root  34250,  we  find  185;  hence  the  root  required  is  — -,  or  1.85,  a  result 

which  is  within  — -  of  the  true  value. 

It  appears  from  the  above  that  the  number  of  decimal  places  must  always 
be  made  even  before  the  operation  commences. 

If  we  wish  to  have  a  greater  number  of  decimal  places  in  the  root,  we  must 
add  on  the  right  of  34250  twice  as  many  zeros  as  we  wish  to  have  additional 
decimal  figures. 

We  thus  deduce  for  the  extraction  of  the  square  root  of  a  decimal  fraction 
the  following 

RULE. 

Annex  ciphers  till  there  arc  twice  as  many  decimal  places  as  are  required  in 
the  root,  and  then*proceed  as  in  whole  numbers  ;  or,  beginning  at  the  decimal 
point,  point  off  both  ways  the  usual  periods  of  tivo  figures  each. 

By  which  we  obtain 

V 0799. 6516=82. 46,  V?3  .5=8.5,  .y/Tl)T=2. 81. 

EXTRACTION  OF  THE  CUBE  ROOT  OF  NUMBERS. 

i)i.  The  numbers  ' 

1,  2,   3,    4,     5,       6,      7,      8,      9,       10,        100,  1000, 

when  cubed,  become 

1,  8,  27,  64,  125,  216,  343,  512,  729,  1000,  1000000,  1000000000 
and,  reciprocally,  the  numbers  in  the  first  line  are  the  cube  roots  of  the  nuiu 
bers  in  the  second. 

Upon  inspecting  the  two  lines,  we  perceive  that,  among  tho  number- 
pressed  by  one,  two,  or  three  figures,  there  are  only  nine  which  are  perfect 
cubes  ;  consequently,  tho  cubo  root  of  all  the  rest  mast  be  :i  whole  number  plus 
u  fraction. 

92.  But  we  can  prove,  in  tho  same  manner  as  in  the  case  of  tho  squarn 
root,  that  the  cube  root  of  a  whole  number,  which  is  not  the  /"  rfeel  ruin-  of  I 
other  whole  number,  run  not  be  expressed  by  an  cruet  fraction,  and, 

ijin  nil y,  its  cube  root  is  incomnu  nsurable  with  unity. 

93.  The  difference  between  the  cubes  of  two  consecutive  whole  numbers 
reater  in  proportion  as  the  numbers  themselves  are  greater;  tho  expression 

for  this  difference  can  easily  be  found. 


Let 

Then, 
Hence, 


EXTRACTION  OF  THE  CUBE  ROOT.  9\ 

a  and  a-\-l  be  two  consecutive  whole  numbers , 
(a+l)3=a3+3a24-3a4-l ; 


(a  +  l)3— a3=3a2+3a+l ; 
that  is  to  say,  the  difference  of  the  cubes  of  two  consecutive  whole  numbers  it 
equal  to  three  times  the  square  of  the  less  of  the  two  numbers,  plus  three  timet 
tiie  si?nple  poiver  of  the  number,  plus  unity. 

Thus,  the  difference  between  the  cube  of  90  and  the  cube  of  89  is  equal  to 
3X(89)2+3X  89  +  1=24031. 

Let  us  now  proceed  to  investigate  a  process  for  the  extraction  of  the  cube 
root  of  any  number. 

EXTRACTION  OP  THE  CUBE  ROOT. 

94.  The  cube  root  of  a  proposed  number,  consisting  of  one,  two,  or  three 
figures  only,  will  be  found  immediately  by  inspecting  the  cubes  of  the  first 
nine  numbers  in  (Art.  91).  Thus,  the  cube  root  of  125  is  5,  and  the  cube  root 
of  54  is  3  plus  a  fraction,  for  3  X  3  X  3=27,  and  4x4  X  4=64  ;  therefore  3  is 
the  approximate  cube  root  of  54,  within  one  unit  of  the  true  value. 

For  the  purpose,  of  investigating  a  new  and  simple  rule  for  the  extraction  of 
the  cube  root,  it  will  be  necessaiy  to  attend  to  the  composition  of  a  complete 
power  of  the  third  degree.     Now,  since  we  have 

(a4-o)3=(a  +  o)(a  +  o)(a  +  Z>)=a3+3a2o+3aZ>2+Z<3, 
it  is  obvious  that  the  cube  of  a  number,  consisting  of  tens  and  units,  will  be  al- 
gebraically indicated  by  the  polynomial 

a3-f3a26+3aZ>2+o3, 
where  a  designates  the  number  of  tens,  and  o  the  number  of  units  in  the  root 
sought.     The  number  in  the  tens'  place  will  evidently  be  found  by  extracting 
the  cube  root  of  the  monomial  a3,  for  3/a:!=ra,  and  removing  a?  from  the  poly 
nomial  <?3+3a26-r-3a62-r-i3,  we  have  the  remainder, 

3a25  +  3a62+63=(3a2+3ct64-62)6  ; 
and  the  difficulty  that  has  been  hitherto  experienced  in  the  extraction  of  the 
cube  root  entirely  consists  in  the  composition  of  the  expression  Sfi^-j-Sai-f-o'. 
which  is  obviously  the  true  divisor  by  which  to  divide  the  remainder,  aftei 
subtracting  a3,  or  the  cube  of  the  tens,  for  the  determination  of  o,  the  figure 
of  the  root  in  the  place  of  units.  The  part  3a2  of  the  expression  3a24-  3<z&+o2, 
being  independent  of  b,  the  yet  unknown  part  of  the  root,  is  employed  as  a 
trial  divisor  for  the  determination  of  6  ;  but  since  the  expression  3a2-{-3ab-\-b* 
involves  the  unknown  part  of  the  root  in  its  composition,  it  is  obvious  that  the 
trial  divisor  3a2,  which  does  not  contain  6,  will,  at  the  first  step  of  the  opera- 
tion, give  no  certain  indication  of  the  next  figure  of  the  root,  imless  the  figure 
denoted  by  b  be  very  small  in  comparison  with  that  denoted  by  a ;  for  the 
trial  divisor  3a2  will  be  considerably  augmented  by  the  addend  3ab-{-b"  when 
6  is  a  large  number,  while  the  augmentation,  when  6  is  a  small  number,  will 
not  so  materially  affect  the  trial  divisor. 

When  the  figure  in  the  tens'  place  is  a  small  number,  as  1  or  2,  it  is  hence 
obvious  that  little  or  no  dependence  can  be  placed  on  the  trial  divisor;  but  if  a 


9-2  ALGEBRA. 

reat  and  b  small,  the  trial  divisor,  3a8.  will  generally  point  out  the  value 
of  l>.  All  this  will  be  evident  if  we  consider  that  the  relative  values  of  a  and 
ft  materially  affect  the  true  divisor,  3a*-\-3ab-\-b*.  In  the  successive  steps, 
li  >wever,  of  the  cube  rool  this  uncertainty  diminishes;  for,  conceiving  a  to 
designate  a  number  consisting  of  tens  and  hundreds,  and  b  the  number  o 
units,  then  the  value  of  b  being  small  in  comparison  with  a,  the  amount  of  the 
effect  of  b  in  ilia  addend  '.jab-\-b-  will  be  very  inconsiderable  ;  hence  the  trial 
divisor,  3a5,  will  generally  indicate  the  next  figure  in  the  root. 

To  remove,  iu  some  measure,  the  difficulty  which  has  hitherto  been  ex 
perienced  in  the  extraction  of  the  cube  root,  we  shall  proceed  to  point  out  two 
methods  of  composing  the  true  divisor,  3a--\-3ab-{-b -'.  and  leave  the  student 
to  select  that  which  ho  conceives  to  possess  the  greater  facility  of  operation.* 

95.  First  method  of  composition  of3a'2-\-3ab-\-b'1. 
axa         =  a2  a?+3a-b-{-3ab-+b*  (<z  +  b=  root  sought. 

a  a-  X  a  = «3 

a  a2 

—  3a26  +  3ai2+i3 

3a2 
(3a+h)xb=  3aZ>-f-   62 

b  

b  (3a2+3a5+   b-)  X  b  =  . . . .  3a-b+3ab2+b* 

b*  


3a+3fc  3a2-f  Gai-r-C//-'. 

Distinguishing  the  three  columns  from  left  to  right  by  first,  second,  and 
third  columns,  we  writo  a  in  the  root,  and  also  three  times  vertically  in  the 
first  column;  then  aXa  produces  a-,  which  write,  also,  three  times  vertically 
in  the  second  column;  multiply  the  second  a?  by  a,  placing  the  product,  «3, 
under  a3  in  the  third  column  ;  then,  subtracting  a3  from  the  proposed  quantity, 
we  have  the  remainder,  ?M-b-\-3oJ>'-\-}>'.  The  sum  of  the  three  quantities  in 
the  second  column  gives  3a2  for  the  trial  divisor,  by  v, '  i  !i  find  b,  the  next 
figure  of  the  root,  and  to  3</,  the  sum  of  the  last  three  written  quantities  in 
the  first  column,  annex  b  ;  then  the  sum,  3a-{-b,  is  multiplied  by  b,  and  the 
product,  3ab-\-b2,  is  placed  m  the  second  column;  thru  the  trial  divisor. 
and  ihe  addend,  3a&+69,  being  collected,  give  tbe  true  divisor,  3a9-f-3afr-|-&*i 
which  multiply  by  b,  and  place  the  product,  :\,,:b-\-3(tli:-\-b:\  under  the  re- 
mainder, 3a-b-\- 3aos '-\-l}\  When  there  is  a  remainder  after  this  operation, 
the  process  may  bo  continued  by  writing  b  twice  in  the  first  column,  under 
?„i-\-b,  and  h'-  once  m  the  second  column,  under  the  last  true  divisor  ;  thei 
_J_ r;,,/,_i_  ,*)/,-,  the  sum  of  the  last  written  three  lines  in  the  second  column,  will 
bo  another  trial  divisor,  with  which  procoed  as  above.  We  have  written  a- 
in  the  second  column  three  times  in  succession,  to  assimilate  the  first  step  in 
the  operation  1o  the  other  successive  steps,  hit  the  fust  trial  divisor,  :\n:.  may 
lie  written  at  once,  ami  the  symmetry  of  the  disposition  of  the  quantities  in 
the  Brsi  steps  disregarded.! 

*  These  methods  may  be  nnsscd  over  by  the  itadent,  u  will  as  that     Lven  for  tliu  bi- 
i !  the  in, -ill,., I  employed,  which  is  .1.  jcribed  at  (Art  112),  which  ia  appli 
naMe  t"  i  In'  extraction  of  the  root  of  the  third  and  fourth,  ai  well  aa  ofanj  other 
i  Three  quantities  are  added  each  ti ;  in  die  method  on  ne:  two. 


EXTRACTION  OF  THE  CUBE  ROOT.  93 

96.  Second  method  of  composing  3a"-{-3ab-\-b2,  the  true  aivisor. 


a 

a2 

a3+3a2Z>  +  3a&2+&3  (a+6 
.   a3 

3a26  +  3a£3+Z>3 

a 

3a% 

3a  +  b  .  .  . 

3ab+  i2 

tm 

b 

3a2+3a&  +  6s  .  .  . 
3aZ>+2&3 

f?/72_l_fW)_]_.r>,7|2 Rfi 

3a26  +  3aZ>2+&3 

3«+2&  .  .  . 
b 

nnnd  trinl  rlivisnr. 

3a+3& 
In  this  method  we  write  a  under  a  in  the  first  column,  and  the  sum  2a 
being  multiplied  by  a,  gives  2a3  to  place  under  a2  in  the  second  column,  and 
the  sum  of  2a2  and  a-  is  3a2  for  the  trial  divisor.  Again,  under  2a  in  the  first 
column  write  a,  and  the  sum  of  2a  and  a  gives  3a.  Now,  having  found  b  by 
the  trial  divisor,  annex  it  to  3a  in  the  first  column,  making  3a-\-b,  which,  mul- 
tiplied by  b,  and  the  product  placed  in  the  second  column,  gives,  by  addition, 
the  true  divisor,  3a2+3a&  +  &2,  as  before.  We  shall  exhibit  the  operation  of 
extracting  the  cube  root  by  both  these  methods. 
t 

EXAMPLES. 

(1)  What  is  the  cube  root  of  xs— 9a*+39:c*— 99a«+15Ga:s— I44.r+G4? 

By  the  first  method, 
afl  x*  afi—  9x5+39x4— 99x3+156x2— 144x+64  (x—  3x-H 

#2  x* .X6 

X*  X* 

3x* 

3^2 — 3x  .  .      —  9x3+  9x2 


—9x5+39x4—  99x-3 


— 3x       3xi—  9x3+  9x"     ....    —9x5+27x4—27x3 

_3x 12xi— 72x3+156x2— 144x+64 

3x4— 18x3 +27a;2 


3^3 — 9x+4  .  .  .  12j2— 36a+16 

3^4 — 18x3+39x2— 36x+ 16  .  .  .  12*4— 72*3.1.  156*2— 14J3+64. 

(2)  What  is  the  cube  root  of  x6+6^— 40r*+96a:— 64  ? 

By  the  second  method. 
x2  x6+6x5— 40x3-f-96x— 64  (x2-f  2a;— 4 

X2  X* X6 


2.r» 2ar*  6x5— 40x3 

X2  

3_£-t 

3x3+2x  ...  6x3+  4x2 


2x 


3x»+  6x3+  4x2    ...    .        6x5+12x4+  8x3 


3x2+4x     .  .  6x3+  8x2 


2a;  — 12aH — 48x3+96x— 64 

3x4+12x3+12x2 


3x2+6.1— 4  .  .  — 12x2— 24x+lG 

3x4+12x3—  24x+16  — 12.r4— 48x3+96x— 64. 


94  ALGEBRA. 

(3)  What  is  the  cube  root  of  a3+3a=i  +  3aZ;0+63+3a-c4-6a6c+3fc3c+3ac' 
+  36c2+c3?  Ans.  a  +  b  +  r- 

(4)  Extract  the  cube  root  of  &—Gx!i+15x*—20x3-\-15x-—6x-\-l. 

Ans.  x-— 2x-r-l. 

97.  The  same  process  is  employed  in  the  extraction  of  the  cube  root  of 
numbers,  as  in  the  subsequent  examples. 

EXAMPLES. 

(1)  Extract  the  cube  root  of  403583419. 

•  • 

7 49  403583419  (739  =  root 

7  49 343 

7  49  

60583 


147 
213 639 


3  

3  15339 46017 

9  


15987 


1456G419 


2199 19791 


1618491  14566419. 

(2)  What  is  the  cube  root  of  115501303  ? 

•  ■  • 

115501303  (487  =  root 
4 16 64 


4  51501 


3 32 

4  48 


128 1024 

8  

5824 46592 

136 1088  


8  4909303 

—  6912 


1447 1012  I 


701329  4909303. 


98.  The  local  values  of  the  figures  in  the  rool  determine  the  arrangement 
of  tin'  figures  in  the  several  columns,  as  i>  exemplified  by  working  the  la 
ample  as  on  nexl  page;  by  omitting  the  terminal  ciphers,  the  arrangement  is 
precisely  the  same  as  m  the  preceding  example. 


EXTRACTION  OF  THE  FOURTH  ROOT.  9S 

115501303  (400+80  +  7 

400 160000 64000000 

400  

51501303 

800 320000 

400  

480000 

1200 
80 


1280 102400 

80  

582400  46592000 

1360 108800  

80  4909303 

691200 

1440 
7 

1447 10129 


701329  4909303 


99.  Extraction  of  the  fourth  root  of  whole  numbers. 

1  he  investigation  of  a  method  for  extracting  tho  fourth  root  of  any  numbei 
is  similar  to  that  employed  for  the  cube  root.     Thus,  since 

(cr+ &)4=a4+  4a?b  +  6aW+ 4a&3+ b\ 
we  may  conceive  a  to  denote  the  number  of  tens,  and  b  the  number  of  units 
in  the  root  of  the  number  expressed  by  a4+4a3&+6a262+4a&3+&4.  Then 
i/ai=a,  the  figure  in  the  tens'  place,  and  the  remainder,  when  a4  is  removed,  is 
4a36+6«2i2+4a63+64=(4a3+6a2&+4a62+£3)i. 
The  method  of  composing  the  divisor  4a3+6a26  +  4a2>2+&3,  for  the  deter- 
mination of  b,  the  figure  in  the  units'  place,  may  be  illustrated  as  follows : 

ax  a    =  a2  a4+4a3Z>  +  6a2&2+4afc3+Z>4  (a+6 

a  a'X«  =  a3 


2a  x  a    =2a2  a?xa  =a4 


3a2  X  a  =3a3  4a36+6a2£2+4a&3+64 


3a  X  a    =3a2  4a3 

a  


6a* 


(4a+6)6=4a&+62 


(6a2+4a&+62)&=6a2i+4at2+&3 


(4a3+6a26  +  4a&2+&3)&=4a3Z>  +  6a2Z>2+4a&3+&4. 

100.  From  this  mode  of  composing  the  complete  divisor  we  easily  derira 
the  following  process  for  the  extraction  of  the  fourth  root  of  an  7  number. 


9G  ALGEBRA. 

1   \  \MI>LE. 

What  is  tho  fourth  root  of  1185921  ? 


3X3     =      9 

3                     9X3      = 

27 

6X3     =18 

27X3 

3                   27  X  3     = 

81 

9X3     =27 

108  ..  . 

t 

51 

123X3   =      369 

5769X3  = 

17307 

1185921  (33  =  root 
=     81 


375921 


125307X3   =     375921 

in  tho  same  manner,  tho  student  may  readily  investigate  rules  for  the  m 
traction  of  the  higher  roots  of  numbers,  simply  observing  to  use  an  additionn' 
column  for  each  successive  root. 

101.   To  represent  a  rational  quantity  as  a  surd. 

Let  it  be  required  to  represent  a  in  the  form  of  a  surd  of  the  n\\x  order, 

then,  by  (Art.  63),  the  form  will  bo  V«n.  or  (a")n  5  for  by  raising  a  to  the  ?ith 
power,  and  then  extrasting  the  nth  root  of  tho  nth  power  of  a,  we  must  evi 
dently  revert  to  the  proposed  quantity,  a.     Hence  we  have 

a—  -/«2  =  Vd?  =  Va1  =  !$Tau  =  ^am=  Va° 

1  B     '  —   - 

a  =  {a-)2  =  {ai-)*=(a'sf=(a")m. 
102.  When  tho  given  quantity  is  the  product  of  a  rational  quantity  and  a 
surd,  we  must  represent  the  rational  quantity  in  the  form  of- the  given  surd, 
and  then  express  tho  product  with  a  single  radical  sign,  or  fractional  index 
Tb"S,  we  have 

ay/b   =  -v/""X  Vb=  Va"b 

\/7>Z=  V 3a  X  3a  X  Vltb    =  ■y/9d:Xob    =  -^IbcTb 
ay~xi)=  V  ax  ax  aX  V^/=  V^X  Vxy=  Va*xy 

lJ-v/7   =  VUlX  >/"!  =-/ll*X7     =-v/1008 

i  i 


a{l—a--.L-ys={a:)-  {I—  =  (a9—  =  ■/«*—    . 

(1)  Represent  as  in  the  form  of  a  surd,  whose  index  is  5. 

(2)  Represent  2 —  -/3  in  the  form  of  a  quadratic  snrd. 

(3)  Transform  6  ^11  into  the  form  of  a  quadratic  snrd. 
(1)  Transform  a-)/ a  —  b  into  the  form  of  a  quadratic  surd. 


(5)  Represent  as  a  surd  tho  mixed  quantity  (a "  +  .'/)/' 


•r+.V* 


(6)  Represent  as  a  surd  tin*  mixed  quantity  (r-f-  1)    / 
(I)*/"17,  m*  (fi"  i". 


\  sc+4* 

\N       \\    | 


(2)  V7-4V3- 

(3)  J 


(I)    v'-  —  </•'/«  or  (<r1— fl'o) 
(6)    /^  -;/')* 

.      I  i  or(k+4)J. 


'• 


BINOMIAL  SURDS.  97 

103.   To  find  multipliers  which  will  render  binomial  surds  rational. 
The  product  of  two  irrational  quantities  is,  in  many  instances,  a  rational 
quantity,  and,  therefore,  an  irrational  quantity  may  frequently  be  found,  which, 
employed  as  a  factor  to  multiply  some  other  given  irrational  quantity,  will 
produce  a  rational  result ;  thus, 

y/aX  Va       =a 
yxx  Vx"     =x 

i/yxi/ym-l=y> 

Again,  since  the  product  of  the  sum  and  difference  of  two  quantities  is  equal 
to  the  difference  of  their  squares,  we  have,  evidently, 
(  ya  —  y/b){  ■/«+  y/b)=a  —b 
(x+     yfy){x     -yy)=x"-y 
( Va-'—    y){  V*+    y)  =*  — 2/2- 

Hence  it  is  obvious  that,  in  these  and  similar  equalities,  if  one  of  the  factors 
be  given,  the  other  factor  or  multiplier  is  readily  known,  and  the  proposed 
irrational  quantity  is  thus  rendered  rational.  By  a  double  operation  of  this 
kind,  multiplying  (  V«+  Vp-\-  V'l)  by  ( Vw+  Vl>—  Vq)>  we  have  (  y/n 
_f_  y/py—q,  or  n-\-p  — q+2y/np;  and  multiplying  this  by  n-\-p— q— 2  y/np, 
the  given  expression,  \/n-\-  \/p-\-  V(?>  i3  rationalized.  In  the  same  manner, 
since 

{x  ±  y)  (x~  ^  xy + y")  —x3  i  y* 
.'.  (yx±yy)(yx^¥xy+¥y*)=x±y, 
and  the  expression  Vxi  y/y  may>  therefore,  be  rationalized  by  multiplying  it 
by  fy&^=f/xy+tyy*-,  and  fyx*=f  fyxy -{•  fyy*,  multiplied  by  tyx^yy,  will 
produce  a  rational  result. 

Again,  by  division  [see  Art.  23  (5),  (6),  (7)], 

x  ~~y  —xn-1^-x"-J2y-{-xn-3yi-\-xn~iy3-\- +2/n_1 

x    y 

n — 1        „.n— 5».    I    ^.n — 3172__Tn — *1/3-L  —  I/"-1 


x+y 

xn-\-yn 


=xJ'-1—xn-2y+xa-3y'2—x*-4y3-{-  ....  —  yn 


x+y 


z=xn-1—xn-2y^-xn~3yi — x°— 1y3+  •  •  •  •  +2/n~ 


Put      xn—a  ;  theD  x=  Va ;  xn~]  =  V«n_1 !  zn_2=  V  an~2,  &c. ; 
yn=b ;  then  yz=  \Vb  ;  nf    =  yV ;  y3=  yb3,  &c. ; 
hence,  by  substitution  in  the  three  preceding  equalities,  we  have 

aa~yb=  V^=I+  •v/«^+  V«n-^'2+  Va^M h  Vb^1  •  (1) 

a  —  6 


a+b 


Va"-1—  Va"~2*+  yan'3b2—  Va"^4t3+  •  •  •  —  Vb"'1  ■  (2) 


=  Va0-1—  Van^+  Va^V—  Vaa-*b3^ \-  Vb°~l .  (3) 


V«+  Vb' 

Now,  the  dividend  being  the  product  of  the  divisor  and  quotient,  it  is  obvi- 
ous that  a  binomial  surd  of  the  form  y/a—yb  will  be  rendered  rational  by 
multiplying  it  by  n  terms  of  the  second  side  of  equation  (1),  and  a  binomial 
surd  of  the  form  V a-\-  yb  will  be  rationalized  by  employing  n  terms  of  the 
second  side  of  equality  (2)  or  (3),  according  as  n  is  even  or  odd,  the  product 
in  the  former  case  being  a — b,  and  in  the  latter  a — b  or  a-\-b. 

G 


98  ALGKBKA. 

Note. — When  n  is  an  even  number,  employ  equation  (2),  and  when  it  is  aa 
odd  number,  equation  (3),  in  order  to  rationalize  y  a+  yb. 

EXAMPLES. 

(1)  Find  a  multiplier  to  rationalize  v/11  —  ^/7. 

Employing  equation  (1),  we  have  a  =  ll,  6=7,  and  72=3  ;  hence  required 
multiplier  =^li»+ 101^4-^=^121+^77+^49. 

And,  VIS  +V77+V49 

Vll     -V7 


Vl331+\/847+V539 


_  ^647—  V539—  V343 
11  *  *  —      7     =4,  a  rational  product. 


(2)  Rationalize  the  binomial  surd  -v/5+^/4. 

Hero  we  have  a=5,  6=4,  n=3,  an  odd  number;  hence  by  equation  (:J) 
we  have  multiplier  required,  =  ^25 —  s/20-\-\/liJ;  and,  by  multiplication, 
(^5+ £/4) (^25  —  ^/20  +  s/T6)=5+4  =  9=  a  rational  number. 

(3)  What  multiplier  will  render  the  denominator  of  the  fraction  -^p. — irp, 

a  rational  quantity  ? 

5 

(4)  Change  .. .    -  ,0  into  a  fraction  that  shall  have  a  rational  denominator 

j/x- 

(5)  Change  •-.-  ,  ., ,    ..  „  into  a  fraction  that  shall  have  a  rational  de 

nominator. 


Va+£+  V# — x 
(6)  Change     ,  .  into  a  fraction  that  shall  have  a  rational  de- 


nominator. 

ANSWERS. 


(3)  V74+ V73.2+^/7-.2-+ V7.23+ V24- 
5(yiG+ffS+V4) 

(4)  o • 

(5)  V^Vx^^y)_x^Vl^ 
1  x±y  x^y 

(G) • 

104.   To  extract  the  square  root  of  a  binomial  surd. 

x>eforo  commencing  the  investigation  of  the  formula  for  tin*  extraction  of 

the  square  root  of  a  binomial  surd,  it  will  be  necessary  to  premise  two  or  three 
lemmas. 

Lemma  1.  Tho  square  root  of  a  quantity  can  not  be  partly  rational  and  partly 
irrational. 

For,  if  i/a=b-\-  -y/r,  then,  by  squaring,  we  b 

(1  —  t3  —  c 
a=b--\-c-T-2b  V1' ;  therefore,  ■/.-= — ^r — ; 

that  is,  an  irrutional  equal  to  a  rations!  quantity,  which  is  absurd. 


BINOMIAL  SURDS.  99 

Lemma  2.  If  «i  -y/&=-r=t  *Jy  be  an  equation  consisting  of  rational  and  ir- 
rational quantities,  then  a=x,  and  •;/&=  Vy !  *•  «•>  the  rational  ana  iiTational 
parts  of  the  two  members  of  an  equation  must  be  separately  equal. 
For,  if  a  be  not  equal  to  .r,  let  a — x  =  d ;  then  we  have 

i  Vy-f-  Vb=a — x;  but  a — x=.d;  therefore 

i  Vz/T  Vb=d,  which  is  impossible  ; 

.•.  a=x,  and,  taking  away  these  equals,   -\/b  =  -yjy. 


Lemma  3.  If  -/«+  V^=;r+1/)  then    V 'a — *Jb=x — iy  ;  where  r  and  y 
are  supposed  to  be  one  or  both  irrational  quantities. 

For,  since  a-\-  iJb=x--\-y"-\-2xy ;  and  since  x2  and  y"  are  both  rational, 
2xy  must  be  irrational.     By  Lemma  2,  we  have 

a=x2+2/2;    ^b=2xy 
.•.  a —  *Jb=X" — 2xy-\-y* 

and  -\/a —  -\/b=x — y. 

Let  it  now  be  required  to  extract  the  square  root  of  a-\-  -y/6. 

Assume  Va-\-  Vb=x-{-y  ;  then  ^J a — <\/b=x — y 

.'.  a-{-  -\/b=x"-\-y:2-\-2xy 
a —  -\/b=x'2-\-y'2r — 2xy 

.-.  By  addition,  2a  =2(x"+y"),  or  a=x"+y*. 

Again,  y/~a~+V~bX  Va—  Vb=x°—y'2,  or  y/ai—b=:x'i—y3. 
Hence  x*Jry'2=a 


x" — y~=  V^ — b=c,  suppose. 
Therefore,  by  addition  and  subtraction,  we  have 

a-^-c       ,  a — c 

X2=    2     and  2/2==_2~" 

\a-\-c  la — c 

•••  x=V-2~         y~V~2~' 

. jaJrc       la  — 

Hence  V«+  ^b==\~2~^"\~2 

\a-\-c        la  — 

V«-V^=V~2-~V~2" 


!  —  C 


! C 


(1) 


(2) 


where  c=  V^2 — &  »  and>  therefore,  a2 — Z»  must  be  a  perfect  square  ;  and  this 
is  the  test  by  which  we  discover  the  possibility  of  the  operation  proposed.* 

*  When  the  quantity  a2 — b  is  not  a  square,  the  values  of  a  and  b  are  no  longer  rational 
but  it  is  clear  that  the  formulas  (1)  and  (2)  will  still  give  true  results.     As,  howe7er,  these 
are  more  complicated  than  the  original  expressions  themselves,  they  are  rarely  employed 
yet,  when  -\/b  is  imaginary,  the  result  merits  attention. 

In  order  to  examine  this  case,  change  b  into  — b";  a-\-\/b  becomes  a-\-b\/ — 1.  The  re- 
markable circumstance  just  alluded  to  is  this,  that  the  square  root  of  a-\-b\/ — 1  has  tho 
same  form  as  this  quantity  itself. 

This  is  shown  from  the  formula  (1),  for  since  c=\/ a*-\-b'2,  when  b  is  changed  into  — #», 


the  second  member  becomes     Ja-\--[/a--i-b:^_    la     \Za--\-b~^     The  quantity  under  the 
first  radical  is  positive,  and  that  under  the  second  negative,  since  -\/a--\-b-  is  greater  thai 


100  ALGEBRA. 

EXAMPLES. 

(1)  "What  is  the  square  root  of  11+  \f72,  or  11 +  G  y/2  ? 

Here              c  =  ll ;  b=72;  c=  Va2— 6  =  Vl^l —  72=7 
. —        /a-l-c        /a — c 

.-.  7n+(i  v~,=v^:-+v^_=3+  ^~- 

(2)  What  is  the  square  root  of  23— 8  -y/7  ? 


Here      a=23  ;   b  =  82X  7=448;  c=  -/a5— b  =  Vo29  —  448=9 


. r-        /a+c        /a — c 

V23-8V7=j^--yJ—=4-V7. 


(3)  What  is  the  square  root  ofl4  +  6-/5?  Ans.  3  +  y/'o 

(4)  What  is  the  square  root  of  18 ±2  y/T7  ?  Ans.  -/lT±  V? 

(5)  What  is  the  square  root  of  94  +  42  y/5  ?  Ans.  7  +  3  V 5 

(6)  To  what  is  '\np-\-2m2 — 2my/np-\-mi  equal  ?  Ans.   y/np-\-7nr — nt 

(7)  Simplify  the  expression\/ 16+30  -y/  —  1+V16— 30  V  —  1-      Ans.  if». 

(8)  To  what  is  •v/28+10-/3  equal?  Ans.  5+  y/Z. 

(9)  \lbc+2by/bc—b2—\Jbc—2bi/bc  —  b'i=±2b 


v/ 


(10)  Vafc+4c2— ^+2  V4a6c-  —  «W-=  ■/<*&+  •v/4c:!— c?3. 

(11)  What  is  the  square  root  of — 2y/ — 1?  Ans.  1 — \/ — 1. 

(12)  What  is  the  square  root  of  3  — 4  ■/— 1  ?  Ans.  2—  V  -1 

„„       .  3-/3  +  2-V/0    112+20 -/T^, 

(13)  What  is  square  root  of  — — J~ —  . /3 —  * 

Ans.  (1+  y/2)  •  (5+  y/3) 


BINOMIAL  THEOREM. 


105.  It  is  manifest,  from  what  has  been  said  above,  that  algebraic  polynomials 
may  be  raised  to  any  power  merely  by  applying  t ho  rules  of  multiplication. 
We  can,  however,  in  all  cases  obtain  the  desired  result  without  having  rocourse. 
to  this  operation,  which  would  frequently  prove  exceedingly  tedious.  When 
a  binomial  quantity  of  the  form  x+a  is  raised  to  any  power,  the  successive 
terms  are  found  in  all  cases  to  bear  a  certain  relation  to  each  other.  This  law, 
when  expressed  generally  in  algebraic  language,  constitutes  what  is  called  the 
"Binomial  Theorem."  It  was  discovered  by  Sir  Isaac  Newton,  who  seems 
to  have  arrived  at  the  general  principle  by  examining  the  results  of  actual  mul- 
tiplication in  a  variety  of  particular  cases,  B  method  which  wo  shall  here  pursue, 
and  give  a  rigorous  demonstration  of  the  proposition  in  a  subsequent  article  of 
this  treatise. 

a;  representing  the  quantity  under  tho  first  radical  by  a8,  and  that  under  tho  second 
by  — /i-,  tho  expression  takes  tho  form  a+/?\/ — 1 ;  hence 

v/<i+V— l=a+/V— l- 

a  E.  D 


BINOMIAL  THEOREM. 


101 


Let  us  form  the  successive  powers  of  x-\-a  by  actual  multiplication. 
z  -\-a 
r  -{-a 
r2-f-  x  a 

-f-  x  a-\-a2 

r2-f-2x  a-\-a2 2d  power. 

r  -f-   a 
r*+2x'-a+     x~a? 

-{-  x2a+  2x0?+ a? 

./'-(- ;;./-ci+  ;:.v(i:-\-,r- 3d  power 

x  -}-  a 

x4+3xia+  3.r-a--i-     ia3 

-f-  x3g+  3x2q24.  3xa3-fa4 

x4+4x3a-f-  6x2a2+  4xa3-|-a4 4th  power. 

x  +   a 

x-"*+4.r4a+  Cx^-j-  4x2a3-f     ^ 

+  x4a+  4x3a2-|-   6x2a3+  4xa4+a5 
i^+5.r»a+10r!a2+10a-2a34-  5xa4+a5 5th  power 

J+   «» 

x« + 5x*a +  lOx'a2  -f- 1  OxW  -f  5x2a4  +     x~aJ 
+  r^a^-  5r*a24-10x3a3+10.r2a4-r-  5xa5+a6 

c0+6x5a  +  15x4a2+20x3a3-j-15x2a4+  6xa5+a6 6th  powei 

x  -\-  a 

x7+6x6a+15x5a2+20x4a34-15x3a44-  6x2a5-|-  xa6 

-f-  .t^a-f-  6.r5a24-15.r4a34-20.r3a44-15.r2a54-6.ra64-a7 
x7+ 7xGa+ 21x5a2+ 35x4a3+ 35x;a4-f-  21x2a5+ 7xa6-fa7  ....  7th  power. 

In  order  that  these  results  may  be  more  clearly  exhibited  to  the  eye,  we 
shall  arrange  them  in  a  table. 

TAJSLE   OF   THE   POWERS   OF  X-\-d. 


(x+a) 


(x  +  ay 


x2+2xa  +  a2 


x-f-a 


(x+a)= 


x3-|-3x2a  +  3xa2+a3 


(x+a)4 


r*-\-  4x3a+  6x2a2+  4x«3  +  a4 


(x+ay 


x5+5x4a+10x3a2+10x2a3+   5xa4  -fa5 


(x+a)< 


xfi+6xr,a+15.r4a24-20.r3a3+15x2a4+   6xa5  +  a6 


(*+«)' 


x"  +  7xpa+21xsa2+35x4tf34-35x'5a4+21x2a5+   7xa6  +  a7 


(x+a)< 


r8+  8x7a+ 28.r«a24-  56.r5a3+ 70x*a4+ 5Gx3a5+ 28x2a6+  8xa7+ a8. 


In  the  above  table,  the  quantities  in  the  left-hand  column  are  called  the  ex- 
vressions  for  a  binomial  raised  to  thejirst,  second,  third,  &c,  pouter  •  the  cor- 


102  ALGEBRA. 

responding  quantities  in  tho  right-hand  column  are  called  the  expansions,  or 
developments,  of  those  in  the  left. 

10G.  The  developments  of  the  successive  powers  of  x — a  are   prec 
tho  same  with  those  of  x-\-<t,  with  this  difference,  that  the  signs  of  the  terms 
are  alternately  -4-  and  —  ;  thus, 

(r— a)6=x*— Sr'a-J-lOrV  —  WxW+'xca*— a\ 

and  so  for  all  the  others. 

107.  On  considering  the  above  table,  we  shall  perceive  that, 

I.  In  each  case  the  first  term  of  the  expansion  is  the  first  term  of  tho  bi- 
nomial raised  to  the  given  power,  and  the  last  term  of  the  expansion  is  the 
second  term  of  the  binomial  raised  to  the  given  power.  Thus,  in  the  expan- 
sion of  (x-\-ay  the  first  term  is  ar1,  and  the  last  term  is  a*,  and  so  for  all  the 
other  expansions. 

II.  The  quantity  a  does  not  enter  into  the  first  term  of  tho  expansion,  but 
appears  in  the  second  term  with  the  exponent  unity.  The  powers  of  .r  de- 
crease by  unity,  and  the  powers  of  a  increase  by  unity  in  each  successive 
term.  Thus,  in  the  expansion  of  (.'"+")''  wo  have  x6,  a*a.  arte3,  arte3,  x*a1, 
xab,  a6.  ., 

III.  Tho  coefficient  of  the  first  term  is  unity,  and  the  coefficient  of  the 
second  term  is,  in  every  case,  the  exponent  of  the  power  to  which  the  binomial 
is  to  be  raised.  Thus,  the  coefficient  of  the  second  term  of  (x-^-a)*  is  2,  of 
(x+cf)6is  G,  of  (x+a)7is  7. 

IV.  The  coefficient  of  any  term  after  the  second  may  be  found  by  multiply 
mg  the  coefficient  of  the  preceding  term  by  the  index  of  .r  in  that  term,  and 
dividing  by  the  number  of  terms  preceding  the  required  term.  Thus,  in  the 
expansion  of  (x-\-a)4  the  coefficient  of  the  second  term  is  4  ;  this  multiplied 
by  3,  the  index  of  .r  in  that  term,  gives  12,  which,  when  divided  by  "2,  the  num- 
ber of  terms  preceding  the  third  term,  gives  G,  the  coefficient  of  the  third  term. 
Again,  6,  the  coefficient  of  tho  third  term  multiplied  by  2,  the  exponent  of  x 
in  that  term,  gives  12,  which,  when  divided  by  3,  the  number  of  terms  pre- 
ceding the  fourth  term,  gives  I.  the  coefficient  of  the  fourth  term.  So,  also, 
35,  the  coefficient  of  the  fifth  term  in  the  expansion  of  (.r-4-(r)7,  when  multi- 
plied by  3,  the  index  of  X  in  that  term,  pives  105,  which,  when  divided  by  5, 
tho  number  of  terms  preceding  tho  sixth,  gives  21,  the  coefficient  of  that 
term. 

By  attending  to  the  above  observations  wo  can  always  raise  a  binomial  of 
the  form  (r-fa)  to  any  required  power,  without  the  process  of  actual  multi- 
plication. 

EXAMPLK   I. 

Raise  x-\-a  to  tho  9th  power. 

The  first  term  is 2*0°, 

The  second  term  is      .     •  9x*al ; 

9X8 
The  third  term  is   ...  .  — — x7a9=  3Gx7a* ; 

The  fourth  term  w     ...  — — xca:{=  8ir*<P, 

J 


BINOMIAL  THEOREM.  103 

The  fifth  term  is — : — x5a*=\26x6a* ; 

4 

m,       •    ,  •  126X5 

The  sixth  term  is — - — r*a5=126.r4a5 ; 

o 

™,  ,  126X4 

The  seventh  term  is — - — x3aG=  8i3?a°; 

o 


The  eighth  term  is — ^—  x2a7=  36x2aT ; 


84X3 
7 


36X2 

The  ninth  term  is — - — x1a8=     9x1a9 ; 

8 

9X1 
The  tenth  tenn  is — - — x°a9=       x°a». 

Hence, 

(x+a)9  =  x9+9x3a+36x7a2  +  84x5a3  +  126x5a4+126x1a5+84x3a6  +  36xW 
+  9xa8+a9. 

EXAMPLE   II. 

In  liko  manner, 

(x— a)w=xw— 10x9a+45x8a2— 120x7a3+210x6a*— 252x5as+210x,a6— 120 
a?a? + 45x2a8 — 1  Oxa9 + a10. 

08.  The  labor  of  determining  the  coefficients  may  be  much  abridged  by 
attending  to  the  following  additional  considerations : 

V.  The  number  of  terms  in  the  expanded  binomial  is  always  greater  by 
unity  than  the  index  of  the  binomial.  Thus,  the  number  of  terms  in  (x+a)4 
is  4  +  1,  or5;  in  (x+a)10  is  10  +  1,  or  11. 

VI.  Hence,  when  the  exponent  is  an  even  number,  tho  number  of  terms  in 
the  expansion  will  be  odd,  and  it  will  be  observed,  on  examining  the  examples 
already  given,  that  after  Ave  pass  the  middle  term  the  coefficients  are  repeated 
in  a  reverse  order  ;  thus, 

The  coefficients  of  (x+a)4  are  1,  4,  6,  4,  1. 

The  coefficients  of  (x+a)6  are  1,  6,  15,  20,  15,  6,  1. 

The  coefficients  of  (x+a)3  are  1,  8,  28,  56,  70,  56,  28,  8,  1. 

VII.  When  the  exponent  is  an  odd  number,  the  number  of  terms  in  the 
expansion  will  be  even,  and  there  will  be  two  middle  terms,  or  two  contiguous 
terms,  each  of  which  is  equallj*  distant  from  the  corresponding  extremities  of 
the  series ;  in  this  case  the  coefficient  of  the  two  middle  terms  is  the  same, 
and  then  the  coefficients  of  the  preceding  terms  are  reproduced  in  a  reverse 
order;  thus, 

The  coefficients  of  (x+a)3  are  1,  3,  3,  1. 

The  coefficients  of  (x+a)5  are  1,  5,  10,  10,  5,  1. 

The  coefficients  of  (x+a)7  are  1,  7,  21,  35,  35,  21,  7,  1. 

The  coefficients  of  (x+a)9  are  1,  9,  36,  84,  126,  126,  84,  36,  9,  1. 

109.  If  the  terms  of  the  given  binomial  be  affected  with  coefficients  or  ex- 
ponents, they  must  be  raised  to  the  required  powers,  according  to  the  princi- 
ples already  established  for  the  involution  of  monomials  ;  thus. 


104  ALGEBRA. 


EXAMPLE  III.' 


Raise  (2r,-f-5a2)  to  the  4th  power. 

The  first  terra  will  be (2r>)*  =l€x"  ; 

The  second  terra  will  he  ....  4(2.r1)3X  (5a2)  =4x8x5iV, 

4X3      , 
Tie  third  term  will  be -^— x  (^x  (5a:)-=Gx4  x25xV», 

6X2 
The  fourth  term  will  be    ....  -^-(2x3)1  X  (5a2)3     =1  X2X  ISor'a9, 

4 
The  fifth  terra  will  be j^x3)0  x  (5a"-)<  =  G25a8; 

.-.  (2r3+5a2)4=16x12+160x9a2  +  600x^<+1000x3a64-625a8. 

EXAMPLE  IV.* 

In  like  manner, 
( a3+  3ab )»  =  (a3)9  +  9  (a3)8  X  ( Sab )  +  36  (a3)7  X  ( 3a&  )2  +  84  (a3)6  X  ( 3a5 )» 
+  126 (a3)5  X  (3aZ>)'  +  126(a3)<X  (3a&)5  + 84(a3)3  X  (3a6f 
+  36(a3)2  X  (3a6)7+ 9a3  X  (3a6)8+  (3ai)9 
=a27-f  27oM  6  +  324a23 Zr  -f  2268a21 63  +  10206a19  b*  +  30618a17  o* 
+  61236a15i6+78732a13i7+5,9049aut3+19683a9i9. 
110.  We  shall  now  proceed  to  exhibit  the  binomial  theorem  in  a  general 
form.     Let  it  be  required  to  raise  any  binomial  (r+a)  to  the  power  represent- 
ed by  the  general  algebraic  symbol  n.     Then,  by  the  preceding  principles,  we 
shall  have 

The  first  term xD ; 

The  second  term nx^a  ; 

n(n  —  1) 
The  third  term 0    'x°--a- ; 

n{n  —  l){n—2) 
The  fourth  term Too xn_3a5 ; 

n(n  —  l)(n  —  2)  In— 3) 
The  fifth  term -5 \\~A    lx°-*a*, 

&c &c. 

The  last  term a". 

The  whole  number  of  terms  will  bo  n-\-\,  and  the  coefficients  be  repeated 

/n  +  l\  in        \ 

a  reverse  order  after  the  I — - — 1"',  or  (^+1 )"'  terra,  according  as  n  is  odd 

or  even ;  moreover,  the  terms  will  all  have  the  sign  +•  ''  'I"'  quantity  to  be 
expanded  be  of  tho  form  x-f-a,  and  they  will  have  the  sign  -\-  and  —  alter- 
nately, if  the  quantity  be  of  the  form  x — a.     Hence,  generally, 

n(n  —  1)              n(n—l)(n—2) 
(i-fa)"=.rn+7ixn-1a+^pr- V~2a2+-^ — -'-^ '-xn~  <r+ 

n(n— l)(n— 2)  w(/j  —  l) 

+         ;  .,;> Va"-3+  \       V,r  -•-|_„.m"-1+  a- 

n(n—l) 
(r-rt)"=r"-)i.i»-1a-| — — xn     <r ±a°. 

i  •  - 

•  The  1>'  it  method  of  pi  I  unplei  is  to  r.iise  the  fourth  and 

ninth  powers,  and  then,  in  thi  thua  obtained,  to  snbatitv  lory,  and!         r 

s  in  the  Brat,  and  ■ii  for  >/.  and  Sai  for  t  In  die  leoond. 


ui 


BINOMIAL  THEOREM.  105 

In  this  last  case  £n  be  an  even  number,  the  last  term,  being  one  of  the  odd 
terms,  will  have  tho  sign  -f-  ;  and  if  n  be  an  odd  number,  the  last  term,  being 
one  of  the  even  terms,  will  have  the  sign  — . 

Both  forms  may  be  included  in  one  by  employing  the  double  sign. 

(4«)"=.t»±B^"- 1<       w(7i-l)x„_2a       w(»-l)(n-2)jn_3a3      &^ 

L.2  1.2.3 

If  we  make  x  and  a  each  equal  to  1,  (x-\-a)n  becomes  (l-f-l)n,  or  2n,  and  the  second  mem- 
ber reduces  to  its  coefficients  ;  hence  the  sum  of  the  coefficients  in  the  binomial  formula  is 
equal  to  the  «,b  power  of  2. 

EXAMPLE  V. 

To  exemplify  the  application  of  the  theorem  in  this  form,  let  it  be  required 
to  raise  x-\-a  to  the  power  5. 

Here  we  havo  w=5,  n — 1=4,  n  —  2=3,  &c. 
Hence, 

xn is  2s  =     x5 

nxn~la is  5x*a  =  5x*a 

n(n—l)  5.4 

•  'x"-^2 is  — r¥  ■=10x3a2 

n(n—l)(n—2)  5.4.3 

'     -xn~3a3 is  x2a3         =  10x2a' 

n(n—l)(n—2)(n—3)  5.4.3.2 

1   2.3.4 lx*~4ai is  172-374^       =  ^ 


n(n  —  l)(n— 2)(n  — 3)(n— 4)  5.4.3.2.1 

-J: i± 11 '-± '-x°a5 s x°< 

1.2.3.4.5  91.2.3.4.5^ 

(x+a)5=x5+5x4a+10r5a2+10x3a3+5.ra4+a5 


EXAMPLE  VI. 

Raise  5c2— 2i/z  to  the  4th  power 

Here, 

.•.xa becomes  (5c2)4  =  625c8 

nxa~l  a becomes  4(5c-)3  X  (2yz)  =  l000c*yz 


.r.=5c2 
a=2y: 
«=4 


n(n— 1)  4.3, 

~2a* becomes        p^c2)2  X  (~1/zf=  600c*yh3 

11     ~~>  I  o~~  3;n"'3a3  •  •  -becomes    T1^(5c2)1  X  {^y~f=  IGOc-yV 

n'n-1f2-f''~r')x--%'becomesi||J(5^rx(%:)'=     16^ 

.-.  (5c2— 2yzy=625cs— 1000csyz  +  600cy-z~— lGOc-ifz^+lGy'z*. 

111.  We  have  sometimes  occasion  to  employ  a  particular  term  in  the  ex 
pansion  of  a  binomial,  while  the  remainder  of  the  series  does  not  enter  into  our 
calculations.  Our  labor  will,  in  a  case  like  this,  be  much  abridged,  if  we  can 
at  once  detennine  the  term  sought,  without  reference  either  to  those  which 
precede,  or  to  those  which  follow  it.  This  object  will  be  attained  by  finding 
what  is  called  the  general  term  of  the  series. 

If  we  examine  the  general  formula,  we  shall  soon  perceive  that  a  certain 
relation  subsists  between  the  coefficients  and  exponents  of  each  term  in  the 
expanded  binomial,  and  the  place  of  the  term  in  the  series ;  thus, 


106  ALGEBRA. 

The  first  terra  is 
arn,  ■which  may  be  put  under  the  form  .rn~1+1 ; 

The  second  term  i9 
nx^-ta  =n.rn-2+la2_1 ; 

The  third  term  is 

n(n  —  1)  n(n— 3  +  2) 

xn  "2a2  =— ^—^x^-Wo*-1  • 

1.2  —   1.(3—1)         -a      ' 

The  fourth  term  is 

n(w  —  1)(»—  2)  n(n  —  l)(n— 4  +  2) 

— (^ — — .rn-3a3  =— -E_J,*-M+ifl4  i  • 

1.2.3  1.2.(4  —  1)       X         a       ' 

The  fifth  term  is 
n(n— l)(n  — 2)(n— 3)  n(n— 1)(»— 2)(»— 5+2) 

— — — -X"~*a*  =— '— Vi-W/t5-1  . 

1.2.3.4  *  —  1.2.3.(5  —  1)  a      ' 

The  sixth  term  is 

n(n  —  l)(n  — 2)(n— 3)(n— 4)  n(n— l)(n— 2)(n  — 3)(n— 6+2) 

; — r— - — ; — zn_Ja;,= ; ! — xa~*+lcP~  • 

1.2.3.4.5  1.2.3.4.(6  —  1) 

Observing  the  connection  between  the  numerical  quantities,  it  is  manifest, 
that  if  we  designate  the  place  of  any  term  by  the  general  symbol  p,  the  /»* 
term  is 

n(n  —  l){n— 2)(n— 3)  (n_„  +  o\ 

'     nWd (/!)  jn-p+laP-'- 

This  is  called  the  general  term,  because  by  giving  to_p  the  values  1,  2,  3,  4, 
we  can  obtain  in  succession  the  different  terms  of  the  series  for  (r+a)\ 

EXAMPLE  VII. 

Required  the  7th  term  of  the  expansion  of  (.r+a)12. 
Here  n  =  12  >  .-.  n—  -p+2=7,  n — jp+l=6 

p=  7  I  ^  —  1=6. 

Substituting  these  values  in  the  general  expression,  we  find  that  the  term 
sought  is 

12.11.10.9.8.7 

■= — = — -     .    -    ^a6,*  or  924xeae. 
1  .  2  .  3  .4.5.0 

EXAMPLE  VIII. 

Required  the  5th  term  of  (2c4— 4/t5)9. 

Here  n  =  d,  p=5,  x=2r«,  a  =  -\lr\ 

.-.n— p+2  =  6,  ?*  — ^>  +  l=5,  p  — 1  =  4  ; 

■    9.8.7.6,  ,   , 

.-.  the  5"'  term  js  /1(~t-')"'  X  (4A6)4,  or  126  X  32  X  256c2%s0. 

Since  tho  second  term  of  the  proposed  binomial  has  tho  sign  — ,  all  the 
even  terms  of  tho  expansion  will  have  the  sign  — ,  and  all  the  odd  terms  the 
sign  +  ;  therefore  the  5th  term  is 

+1032192c9°A90. 

i  \  vmi'i.i:  i\. 

Required  the  middle  term  of  the  expansion  of  (x — a)w. 

Since  the  <\|)unent  is  18,  tin-  whole  Dumber  of  terms  will  bo  I?,  and  heui 


Tli>-  operation  hero  to  bo  performed  is  beat  effected  l>_s  canceling  1 1  it-  tutors. 


HIGHER  ROOTS  OF  NUMBERS.  107 

the  middle  term  will  be  the  10th ;  and  since  it  is  an  even  term,  it  will  have  th* 
sign  —  ;  hence  it  will  be 

18.17.16.15.14.13.12.11.10 
-1.2.3.4.5.6.7.8.9^  "-^OAA 

EXAMPLE  X. 

Required  the  third  and  the  last  terms  of  the  expansion  of  [-x-\-2yj7 

21 
Ans.  —  x'nf  and  128y 

TO  EXTRACT  THE  nfi>  ROOT  OF  A  NUMBER. 
112.  The  nth  power  of  10  is  1  with  n  ciphers,  and  the  nth  power  of  any 
number  below  10  must  be  less,  and  can,  therefore,  bo  composed  of  no  more 
than  n  figures.  The  nth  power  of  100  is  1  with  2n  ciphers,  and  the  nlh  power 
of  any  number  between  10  and  100  can  not,  therefore,  contain  more  than  2n 
figures,  nor  less  than  n.  For  a  like  reason,  the  nth  power  of  three  figures  can 
not  contain  more  than  ?,n,  nor  less  than  2n.  That  of  four  figures  can  not  con- 
tain more  than  4ra,  nor  less  than  3n,  &c.  The  nth  root  of  a  number  being  re- 
quired, it  is  evident  from  the  above  that  there  will  be  as  many  figures  in  the 
root  as  there  are  periods  of  n  figures  in  the  given  number,  counting  from  right 
to  left,  and  one  more  if  any  figures  remain  on  the  left.  The  root  may  be 
divided  into  units  and  lens,  and  the  nlh  power  of  it,  or  the  given  number,  will 
be  equal,  according  to  the  Binomial  Theorem,  to  the  nih  power  of  the  tens  plus 
n  times  the  n  —  1  power  of  the  tens  into  the  units  plus  a  number  of  other  terms 
which  need  not  be  considered.  Tens  have  one  cipher  on  the  right,  and  hence 
the  »"'  power  of  tens  has  n  ciphers  on  the  right  ;  the  n  right-hand  significant 
figures,  therefore,  make  no  part  of  the  nth  power  of  the  tens  ;  to  find  the  tens 
of  the  root,  then,  the  na'  root  of  those  figures  which  remain,  after  rejecting  n  on 
the  right,  must  be  sought  by  an  independent  operation  ;  but  if  there  are  more 
than  n  of  these  remaining  figures,  the  tens  of  the  root  are  expressed  by  a 
number  containing  more  than  one  figure,  which  number  may  be  separated  into 
its  units  and  tens,  the  nlb  power  of  the  tens  of  which  does  not  contain  the  n 
significant  figures  on  the  right  of  that  number  upon  which  the  independent 
operation  is  now  performing,  and  in  consequence  these  n  figures  arc  also  re- 
jected. After  rejecting  periods  of  n  figures  successively,  beginning  on  the 
right  until  there  remains  but  one  period  and  part  or  the  whole  of  another 
period  on  the  left,  let  these  be  considered  an  independent  number,  its  root  will 
contain  two  figures,  tens  and  units;  the  ntii  root  of  the  tens  is  to  bo  sought  in 
what  is  left  after  rejecting  the  right-hand  period ;  the  n  —  1  power  of  the 
tens  has  n  —  1  ciphers  on  the  right;  so,  also,  has  any  multiple  of  this,  and, 
therefore,  n  times  the  n —  1  power  of  the  tens  into  the  units;  which  last 
quantity,  therefore,  is  not  to  be  sought  in  the  n  —  1  right-hand  significant 
figures  ;  after  subtracting  the  n,h  power  of  the  tens  just  found,  only  one  figure 
of  the  next  period,  therefore,  is  to  be  placed  on  the  right  of  the  remainder, 
which  is  then  divided  by  n  times  the  n  —  1  power  of  the  tens;  the  quotient 
will  not  bo  exactly  the  units,  for  the  dividend  contains  also  a  part  of  the  other 
terms  of  the  power  of  the  binomial  which  were  not  considered  :  this  quotient 
may  be  greater  than  the  units  of  the  root,  but  never  can  be  less ;  it  must  be 
diminished  till  the  nlh  power  of  the  two  figures  fouu.  is  equal  to  or  less  than 


108  ALGEBRA. 

tlio  independent  number  under  consideration.  Annex  now  to  this  independent 
number  the  next  period  on  the  right  of  it,  and  consider  what  is  thus  obtained 
as  a  new  independent  number;  the  two  figures  of  the  root  already  found  will 
be  the  tens  of  the  root  of  the  new  Dumber;  bringing  down  our  figure  of  the 
right-hand  period  of  it  to  the  remainder  after  subtracting  the  nth  power  of  the 
two  figures  of  the  root  just  found  from  the  first  independent  number,  and 
dividing  by  n  times  the  n  —  1  power  of  the  tens,  now  composed  of  two  figures, 
a  third  figure  of  the  root  is  obtained  ;  proceeding  in  this  manner,  the  entire  root 
of  the  given  number  will  at  length  be  extracted.* 

EXAMPLES. 

(1)  V504321,  2366=8,921.  (3)    ^ 233416517309451. 

(2)  VH64532,  07234.  (4)    !^282429536481.     « 

113.  By  employing  the  binomial  theorem,  wo  can  raise  any  polynomial  to 
any  power,  without  the  process  of  actual  multiplication. 

For  example,  let  it  be  required  to  raise  x-\-a-\-b  to  the  power  4. 
Put 

<*+*>  =y; 

Then, 

(r+a  +  by  =  (z+yy, 

==.r4+4.r3?/-|-6.r2j/2-j-4;r7/3-r-7/4,  or  putting  for  y  its  value, 
=j-»+4.r'(«  +  fc)4-  Gj*(a  +  b)'+ 4x(a  +  b) ;+  {a+by. 
Expanding   (a-\-b)-,    (a-\-by,   (a-f-i)S   by  the  binomial  theorem,   and   per- 
forming the   multiplications   indicated,  we  shall  arrive  at  the  expansion  of 
(r+a  +  by. 

It  is  manifest  that  we  may  apply  a  similar  process  to  any  polynomial. 
The  following  is  a  demonstration  of  a  general  formula  for  the 

POWER    OF    A    POLYNOMIAL. 

In  the  expression 

(a-f-ft+c+rf....)" 

make  x=zb-\-c-\-d . . .  the  above  power  will  be  equal  to  (a-|-.r)m,  and  by  tn*» 
binomial  theorem  the  term  which  contains  an  in  the  development  of  this  may 
be  written 

1.2.3.4 tnx  an.rm-° 

1.2.3...nXl.2.3...(/«— n)*t  ^ 

Making  y=c-\-d...  wo  have  a,m_"=(/»-j-y)m   ";,  and  developing  this  last  powei 
the  term  containing  />"'  may  bo  put  under  the  form 

*  If  there  be  decimal)  in  the  given  number,  ciphers  mual  be  annexed,  if  necessary,  to 
make  exact  periods  ofd<  id  a  principle  similar  to  dial  explained  in  (Art.  90). 

It"  the  index  of  the  rimt  to  be  extracted  I"'  composed  of  factors,  it  can  be  extracted  by 
means  of  the  successive  roots,  the  degrees  of  which  an   i  ■         ■•<!  by  these  factors.    For 

if  the  m"y^(/"'"P  be  required,  m  have  i/a0U|P=a"P,  T/a°P=aP,  and  y  aP=a. 

The  I  bo  extract  roots  of  numbers  of  a  de  ree  higher  than  the  square  is  by  racaua 

of  lo  rarithms. 

t  This  may  be  obtained  from  the  Ordinary  form  of  the  genera]  term  of  the  hinoinial 
formula 

vt{m—  1) (m  —  ;i-f-l>"nrm-n 

1.2.3....n'  ' 

bv  inultii'lyin  •  both  numerator  ami  denominator  by  1 .  „• .  :i  ...  (m — n). 


HIGHER  ROOTS  OF  POLYNOMIALS.  109 

1.2.3.4 {m  — n)  X  b'"ym-n~a/ 

1.2.3...«'Xl.2.3 {m—n—n')' 

It  is  evident  that  if  this  quantity  be  put  in  the  place  of  a:m~n  in  the  ex- 
pression [a],  the  result  will  represent  the   assemblage  of  the  terms  which 
contain  anbn/  in  the  power  of  the  given  polynomial.     This  result,  after  can 
celing  common  factors,  will  bo 

1.2.3.4....mx  anba'ym-n-'" 

1.2.3...»Xl-2.3...n'Xl.2.3...(m-  n  — n')'        *■  * 
Making  z=zd-\- ...  wo  shall  have  ym-a-a'==(c-\-z)m-n-D',  and  the  term  con- 
taining cL"  will  be 

1.2.3..  .(m  —  n—n')  X  cn"zm-n-n'-"" 
1 . 2  . 3 . . . .  n"  X  1  •  2 .  3 . . . .  (m — n — n' — n")  ' 
substituting  this  expression  for  ym-a~a/  in  [b],  we  have 

1  ■  2 . 3 . .  ■  m  X  a°ba'ca"zm-a-n'-r"' 

1.2.3... «X  1-2.3... «' XI -2.3... n"  XI -2.3...  (m— n— n' — n")' 
It  is  evident  now,  without  carrying  the  reasoning  farther,  that  if  V  be  the 
general  term  of  the  development  of 

(a_|_6_j_c-f-i...)m, 
this  term  may  be  represented  thus, 

1.2.3.4 mXanbn'cn"  ... 

=  1.2.3...wX  1 -2.3... n'X  1.2.3...  »"X  •• 
n,  n',  n" . . .  being  any  positive  whole  numbers  at  pleasure,  subjected  only  to 
the  condition  that  their  sum  shall  be  equal  to  m.    So  that  all  the  terms  of  the  re- 
quired development  may  bo  obtained  by  giving  in  this  formula  to  n,  n',  n" . .  . 
all  the  entire  positive  values  which  satisfy  the  condition 

n-\-n'-\-n" . . .  ,=m. 
When  one  of  these  numbers  is  made  zero,  V  takes  an  illusory  form.     If,  for 
example,  n  =0,  the  series  1 . 2 . 3 . . .  n  placed  in  the  denominator  is  nonsensical, 
because  factors  increasing  from  one  will  never  present  us  with  a  factor  zero. 
To.  relieve  this  difficulty,  let  us  recur  to  the  general  term  [a]  in  the  development 

xm 
of  (a-\-x)m,  and  observe  that  the  hypothesis  n=0  reduces  it  to  - — — — -. 

But  the  hypothesis  «=0  ought  to  give  in  this  development  the  term  which 
does  not  contain  a,  and  this  term  is  xm.  Then,  in  order  that  this  term  shall  be 
deduced  from  the  formula  [a],  it  is  sufficient  to  consider  the  series  1 .2.3...n 
as  equivalent  to  1  in  this  particular  case  of  n=0.  The  same  observation 
should  be  extended  to  the  other  series  of  factors  contained  in  the  denominator 
of  V,  and  then  V  will  give,  without  any  exception,  all  the  terms  of  the  power 
of  the  polynomial  a-{-b-\-c-\-,  &c. 

TO  EXTRACT  THE  mP>  ROOT  OF  A  POLYNOMIAL. 

The  problem  is,  having  given  a  polynomial,  P,  which  is  the  in''1  power  oj 
another  polynomial,  p,  to  find  the  latter. 

Let  us  consider  the  two  polynomials  as  arranged  according  to  the  decreas- 
ing exponents  of  some  letter,  x,  and  call  a,b,c, the  unknown  terms  of  the 

root  p.  They  must  be  such  that,  in  raising  a-{-b-\-c. . .  to  the  power  m,  we 
obtain  all  the  terms  which  compose  P.  But  if  we  imagine  that  we  havo 
formed  this  power  by  successive  multi plications,  it  is  clear  that,  in  the  result, 


NO  ALGEBRA. 

the  term  in  which  x  has  the  highest  exponent  is  the  m*  power  of  a  ;  then  wt 
II  know  the  first  U  rm  of  the  root  sought,  p,  by  extracting  the  m1*  root  of  th» 
first  term  of  the  given  nial,  P. 

The  first  term  of  the  rool  being  found,  it  will  be  easy  to  obtain  the  second; 
but  I  prefer  to  show  at  once  how,  when  we  know  several  successive  terms  of 
the  root  setting  out  from  the  first,  we  can  determine  the  term  which  comes 
immediately  after. 

Let  u  represent  the  .sum  of  tho  known  terms,  and  v  that  of  the  unknown, 
then  P  =  (u-}-r)'",  or,  developing, 

P  =um-\-mum-1v+kun'-*u'i+k'um-:<v3-\-,  &c. 

I  have  not  exhibited  the  composition  of  the  coefficients  k,  k'  ..,  this  not 
being  necessary,  as  will  appear.  From  this  equality,  by  subtracting  um  from 
both  the  equals,  we  obtain 

~P—um=mum-1v-\-Jcum  ->-  +  /.- V,:  HP+,  Arc. 

The  first  of  these  equals,  P— um,  is  a  quantity  which  we  can  calculate  by 
forming  the  mth  power  of  the  known  quantity  u,  and  subtracting  it  from  tho 
polynomial  P.  The  second  is  a. sum  of  products,  by  means  of  which  we  can 
easily  assign  the  composition  of  the  first  term  of  the  remainder  P — ?<"',  and, 
consequently,  discover  the  first  term  of  tho  unknown  part,  v. 

First,  if  we  develop  «m_1,  it  is  clear,  by  the  rules  of  multiplication  alone, 
that  the  first  term  of  the  development,  that  is,  the  one  which  contains  r,  with 
the  highest  exponent,  will  be  a"1-1  ;  then,  if  we  cull/  the  fust  term  of  r,  the 
first  term  of  the  product  mum~lv  will  be  mam~f.  By  a  similar  course  of  rea- 
soning, we  perceive  that  the  first  terms  in  the  developments  of  the  other  prod- 
ucts will   bo  respectively  kam--f2,   k'am-3f3, These  terms,   abstraction 

being  made  of  the  coefficients,which  have  no  influence  upon  the  degree  of  x, 
can  be  deduced  from  the  term  mam"lf,  by  suppressing  in  it  one , or  more  fac- 
tors equal  to  a,  and  replacing  them  by  as  many  factors  equal  to  f.  But/being 
of  a  degree  inferior  to  a  with  respect  to  x,  these  changes  can  give  only  terms 
of  a  degree  inferior  to  mam~f.*  Then,  after  having  subtracted  from  the  given 
polynomial  P  the  7/ilh  power  of  the  part  u  of  the  root  alrea  !y  found,  the  first 

term  of  the  remainder  is  equal  to  the  product  of  m  times  .     -  power  m 1  of 

the  first  term  a  of  the  root  by  tho  first  of  those  terms  which  remain  still  to  be 
found.  Therefore,  dividing  the  first  term  of  the  remainder  by  m  times  the 
power  m  —  1  of  the  first  term  of  the  root,  the  quotient  will  be  a  new  term  of 
this  root.  This  conclusion  furnishes  the  means  of  discovering  successively  all 
the  terms  of  the  root  as  soon  as  the  fust  is  known.  To  Jiavc  the  second  term, 
b,  subtract  from  tin  P  the  mA power  of  t  term  of  the 

root,  then  divide  the  first  term  <f  tlie  remainder  by  ma™-1;  to  have  the  thira 
term,  c,  of  the  root,  subtract  from  P  the  m  ;  <  r  of  a-f-h.  then  divide  the  first 
item  of  this  remainder  by  ma'"  ',  ami  so  on. 

If  in  any  part  of  the  process,  the  remainder  being  arranged  according  to  the 
powers  of  x,  its  first  term  is  not  divisible  by  m  times  the  m  —  1  power  of  | 
iir.>t  term  of  the  root,  tin-  given  polj  Qomial  will  no!  have  at,  exact  root  of  the 
degree  m.  , 

Wo  may  arrange  according  to  the  ascending  powers  of  a  letter,  r,  as  was 

'  Thai,  for  example,  if  a  contain  a£,  and/ conb  ■!  m=I,0,  then  a"   ;/wi!l  r<.. 

J     \\  ill  <-<  nil  :i :  ti  .,'  '',  Iltnl  Sm 


FRACTIONAL  POWERS  OF  BINOMIALS.  Ill 

remarked  at  (Art.  80,  III.),  when  treating  of  the  square  root,  and  the  above 
observation  will  undergo  the  same  modification  as  there  stated. 

It  would  be  superfluous  to  speak  of  the  case  where  the  letter  of  arrangement, 
x,  enters,  with  the  same  exponent,  into  several  terms.  The  method  of  proceed- 
ing in  such  a  case  has  been  already  sufficiently  indicated  in  previous  articles. 

EXAMPLES. 

(1)  Extract  the  5th  root  of  32.r5— 80.r»+80.r3— 40.r3+10.r— 1. 

(2)  Extract  the  6th  root  of  729— 291G.r2+48G0.r4— 4320x6-|-2160.t8— 576x"» 

-f-64x12.  Ans.  3— 2Z2. 

tf)  Extract  the  fifth  root  of  ar20+15.r-,6-5.r-14+90.r-12— 60x-104-282r-» 

—  252.r-6  +  505.r-4  —  496.r-2  +  495  —  4G5r2 
-f  275a.-4— 80x6-f-  15Z8— x10. 

Ans.  £-4-|-3—  x2. 
114.  In  the  observations  made  upon  the  expansion  of  (x-\-a)n,  we  have  sup- 
posed n  to  be  a  positive  integer.  The  binomial  theorem,  however,  is  applica- 
ble, whatever  may  be  the  nature  of  the  quantity  n,  •whether  it  be  positive  or 
negative,  integral  or  fractional.*  When  n  is  a  positive  integer,  the  series  con 
sists  of  n-\-l  terms  ;  in  every  other  case  the  series  never  terminates,  and  the 
development  of  (x-\-a)a  constitutes  what  is  called  an  infinite  series. 

Before  proceeding  to  consider  this  extension  of  the  theorem,  we  may  re- 
mark, that  in  all  our  reasonings  with  regard  to  a  quantity  such  as  (x-\-a)'\  we 
may  confine  our  attention  to  the  more  simple  form  (l-fa)n,  to  which  the 
former  may  always  be  reduced.     For, 

(x+a)  =s(l+j) 

...  (x+ay=\x(l+l)Y 

=zn(l-f--)  ,  or  xn(l+w)n,  if  we  put  -=u 

i            a     n{n—  1)  a?     n(n  —  l)(n — 2)  a3 
=x^l+n.-+-1-ir.-;+ n¥73 .-3+ 

n(n-l)(n-2)(n-3)    a4  ) 
1.2.3.4 T4+'  *°"  S  f 

Suppose  n=-,  where  r  and  s  are  any  whole  numbers  whatever, 

I  r 

Then  (x-\-a.y  becomes  (x-\-a)s,  and  substituting  -  for  n  in  the  series, 

-       -/       a\-  * 

(x+a)*=:3?\l+-y 

a     s\s~  )    a"     s\s       )\s     ~)    a3 


r 
s 


=*•(!  +  -•-+     1>2      -a,-r         L2.3  •& 


r 


(^XHG-3)*. 


+ T2T3T4 ?+'  &0- 


*  A  perfectly  rigorous  demonstration  of  the  binomial  theorem  for  any  exponent  what- 
ever, integral  or  fractional,  positive  or  negative,  will  be  found  towards  the  close  of  this 
treatise. 

t  This  expansicu  may  be  obtained  by  substituting,  in  the  general  form  (Art.  110),  1  foi 


<,  and  —  for  a. 

x 


112  ALGEBRA. 

Or,  reduced, 

!         r    a     r(r — s)    a"     r(r — s)(rv-2s)    a* 
W  =r^  +  «*x+l.ii.s*  •P+     1.2.3.S3        x» 

r(r— *)(r-2s)(r-3*)    B»  ? 

+         1.2.3.4.*  •r4+'  S 

The  binomial  theorem,  under  this  form,  is  extensively  employed  in  analysis 
for  developing  algebraic  expressions  in  series. 

EXAMPLE    I. 

Expand  y/x-\-a  in  a  series. 


V*fa=(*W 


=#+;)*• 


Here  r=l,  s=2. 


j.j.l)         1    a     2\2     V    a2     2\2     vfe      V    ^ 
(1  +  2'i"'"     1.2      V+  1.2.3  V 

i(HG-»)(H «.      i 

+  1.2.3.4  'a-4"*" ) 

(  11113 

A  5         la     2X~~2      a2     2X~~2X— 2      a5 
~X'  I  1_*"2  '  x+     1  . 2      '  •r2"'         1.2.3  '  *• 

113  5  *j 

4- . — .  — 4- ( 

^  1.2.3.4  x*^  ) 

_  ±<         1  a         1       a2         1,3        a*         1.3.5 

— *"  }  1  +  2"x— 1.2.4  '^"^1.2. 3.8'  ?_  1.2.3.4. Hi 

a*  } 

«+ 1 

which  last  may  be  derived  at  once  from  [a],  and  put  under  the  form 

l(         la        la-       1.3      a3       1.3.5      a* 

X~  I    +  2  "  x~J7i  '  x~-"^2  .4.6"  Z3"- 2  . 4 .  G .  8  '  x* 

1.3.5.7       a*  i 

+  2.4.6.8.10  "  ?~  *  &C J 

wherb  the  law  of  the  series  is  evident. 

EXAMPLE    II. 

Expand  -/a2— tz2c-  in  a  series. 

i 

y/ir— a-c-  =  {a-— a-c:)- 

—ail—e")1      Here,  r=l,  6=2,  -=— c* 


x 


$   i     r>(A— 0     sG-OG-2) 
=«  Jl_-. ,,+_  _ .  c, — I72T3— •  <* 

12   3'  "'       •    •   •  •  \ 

<         1  1  1.'.  1.3.5  / 

="  J1 -,'-,..'  '-■■.,iTaX6le,-'lVr'i 


FRACTIONAL  POWERS  OP  BTNOMIALS.  113 

EXAMPLE    III. 


Expand  —  in  a  series. 


m 


=mb        (l  +  -j      =     Herer=l,S=-2,-=rs. 

=  -  \  If!         ~~g("~  2~0        * 

b   I    X  —  2  '  b*  "•"  1.2  "  &4 

"aV-^-1/  \~2~2)    £» 

+  "1.2.3  '   68 

-K-1-1)  (-4-g)  (-I-3)  n\  ( 

+ 17273^ •  y  m 

m)  1    —       ~^X  ~^    ff       ~2X  ~2  X  ~~2 

=  6'(:1— 2'F+         TT2         "6*"*""        1.2.3 
3  5 


2X      2X      2X      2     cl6 


15+ .  &c. 


'  6G~  1.2.3.4  '   i8 

wi  c         1    c4      ^jj    c»     1.3.5   c>~  ? 

=  6"|1_2T2  +  2T4'i*~27T76'"P  +  '      C S 

which  last  expression  might  be  derived  immediately  from  formula  [a].     The 
same  remark  will  apply  in  the  following  examples. 


EXAMPLE  IV. 


n 


Expand  ~         —  in  a  series. 
1         y/b^—c^e3 


n  1 

:n(i2— cV)     2 


c2e» 


=nb       (l— jr)      2      Here  r= -l,s=2, -=  — 

nc  1      c2e2  "^V-^"1/       /c2e2y 

=  6  \ 1  +  2  *  ~6^  +  1.2       •        '   \~b*~) 

"  2  \        2  "     l  /  \     '  2~  2  )         /_£l£lV 


1.2.3 


-§(-4-i)(-5-a)(-5-3W  :  > 


c2e2    1.3   c4e4   1.3.5   c6e6 


_n(     1   c8e3    1  .3  c*e* 


2.4.6  *  66 
1  .  3  .  5  .  7  c«e8  ) 

+  2.4.6.8''F+'&C '    *  'I 

II 


114  ALGEBRA. 

EXAMPLE    V. 

Expand  —===.  in  a  series. 
r         V(w3+"5)3 

p  +  q  3 

■■(p+q)(m3+ns)     * 


a 

=7n~*(p  +  q)(l+—)  Here,r=— 3,  «=4, 


a     n" 

x     ?ft3 
3 


3/3  \ 

i£±ll<       _3     n*      ~4V~4~V      /_n_»y 

:      i»*      j  *  —  4  '  m3  +  1.2  '  \m3) 

4\     4        /\     4        /     /n\3  > 

+  1.2.3  *  W) S 

(.P  +  g)  ,  3     ns         3.7     ^io        3.7.11 

m?       }  *  ~"  4  '  wi3  "*"  1 . 2 .  42  m^"-  T" 


2  .  3  .  43 

n18      3.7.11.15  n-°  > 

7n9"^1.2.3.4.44mu_ '    ' \ 

EXAMPLE  VI. 

1  1  C         2x     3x*     4I3         .      > 

EXAMPLE  VII. 

,  „       n£      #  <  -,       3    a*     3     x*     5     x5     7     a*        -      > 
\  )  \         22    c*     25    c*     27    c6     29   c8  S 

EXAMPLE  VIII. 

—-^     J_(,   ,    3     i     3.13         x»  3.13.23  £ 

a+  101    'l.2.aV        103  1 .  2  .  3  .  a1 
,3.13.23.33             x*            ,      .  ) 

T  10*  1.2. 3. 4.  a*1  ( 

EXAMPLE  IX. 

1 


^—*rT'-3i1+io 


x  ,     6x3       6.11.x3   ,  6.11.16.x*         . 
i=l : -+ —  t  <^c. 


(1+x)*  5     5.10     5.10.15     5.10.15.20 

EXAMPLE  X. 

7  2618       x30 

The  eleventh  terra  of  the  series  for  (a3 — .rV  is — —  .  — -. 

'  4782969   a33 

115.   The  binomial  theorem  is  also  employed  to  determine  approximate 
values  of  t  ho  roots  of  numbers. 
In  the  formula 

(x+«)"=xn(14-n \-— '- — - - £«-H )• 

v  ~    '  v   ~       x         1 . 2       «•  1.2.3  x3  ' 


APPROXIMATE  ROOTS  OF  NUMBERS. 


115 


Let  us  put  «=-,  the  expression  becomes  (x+a)r  or  %/ z+a,  and  we  have 


S/x-\-a=  */x(l+-. 

^         *    v  ^r  x  '       im2 


1    a     r\r       /    a?     r\r       /  \r       /    aJ 


1.2.3 


x3 


+ 


r/   .        1    a      1    r— 1    a*     1    r— 1    2r— 1    a3 


r   x 


2r 


r      yr 


3r       x3 


•) 


If  we  wished  to  form  a  new  term,  it  would  manifestly  bo  obtained  by  mul- 

3r     1       _  a 


tiplying  the  fourth  by 


and  — ,  then  changing  the  sign,  and  so  on  for  the 


rest,  the  terms  after  the  first  being  alternately  positive  and  negative. 

This  being  premised,  let  it  be  required  to  extract  the  cube  root  of  31.  The 
greatest  cube  contained  in  31  is  27 ;  in  the  above  formula  let  us  make  r=3 
r=27,  a =4,  and  we  shall  then  have 


V31=  3/27  +  4 

1       4      1 

-3(l  +  3  •  ^-3 

4  16 

T  27       2187  ^ 


1 

3  '  729 

320 


16       1 
+ 


3 
— ,  &c. 


1 

3 


5 
9 


64 


19683 


-,  &c.) 


531441 
The  succeeding  term  will  be  found  by  multiplying 


320 


by 


3r— 1    a 


531441 

2     4  2560 

-.— ,  and  then  changing  the  sign,  which  will  give  us  —  , 

In  like  manner,  we  shall  find  the  next  term  by  multiplying 


4r 


-,  OF 

X 


2560 


43046721 


by 


4r— 1    a   .       .„     ,        „        ,         2560  11       4 

-57"  -x'  lt  Wdl'  therefore»  be  ^^x-x-:^ 


112640 


r,  and  so  on 


15~27~  17433922005' 
for  any  number  of  terms. 

Let  us,  however,  confine  our  attention  to  the  first  five  terms  of  the  series, 
and  reduce  them  to  decimals ;  we  shall  have,  for  the  sum  of  the  additive  terms, 

f  3=3.00000  ^ 


{ 


-  =  0.14815 


320 


=  0.00060 


^  531441 

And  for  the  sum  of  the  subtractive  tenns, 

16 


\   =3.14875. 


2187" 
2560 


-0  00731 
■  3.00006 


:—0.00737. 


43046721" 
Hence 

>/3T=3. 14138 

a  result  which  we  shall  proceed  to  show  is  within  0.00001  of  the  trum. 

116.  When  the  expression  for  a  number  is  expanded  in  a  series  of  terms. 
the  numerical  values  of  which  go  on  decreasing  continually,  we  easily  perceiv** 


116  ALGEBRA. 

that  the  greater  the  number  of  terms  which  we  take,  the  more  nearh/  shall  wo 
approach  to  the  real  value  of  the  proposed  expression.     Such  a  series  ia  < . 
converging.     If  we  suppose  the  terms  of  the  series  alternately  positive  and 
negative,  we  can,  upon  stopping  at  any  particular  term,  determine  preo 
the  degree  of  approximation  at  which  we  have  arrived. 

Let  there  be  a  series  a  —  b-\-c — d-\-e  — f-\-g — h-\-k —  l-\-m comj» 

of  an  indefinite  number  of  terms,  in  which  we  suppose  that  the  quantities  a, 
b,  c,  d  go  on  diminishing  in  succession,  and  let  us  designate  by  N  the  number 
represented  by  this  series,  we  shall  prove  that  the  numerical  value  of  N  lies 
between  any  two  consecutive  sums  of  any  number  of  the  terms  of  the  above  series. 

For  let  us  take  any  two  consecutive  sums, 

a  —  b-\-c — d-\-e—f  and  a  —  b-\-c — d-\-e—f-{-g. 

Upon  considering  the  first  of  these,  we  perceive  that  the  terms  which  fol- 
low —/are  +(& — ^)4"(^ — OH ;  but  smce  tne  series  is  a  decreasing 

one,  the  positive  differences  g — h,  k — I,  &c,  are  all  positive  numbers;  hence 
it  follows  that,  in  order  to  obtain  the  complete  value  of  N,  we  must  add  to  the 
turn  a  —  b-\-c — J-|-e  — /some  positive  number.     Hence 

a  —  b  +  c  —  d+c  — /<N. 

With  regard  to  the  second  sum,  the  terms  which  follow  +g  are  — (h — k), 

—  (I — m), ;  but  the  partial  differences,  h — k,  I — m,  <5cc,  are  positive; 

hence  — (/( — /.),  — {I — m) ,  are  all  negative,  and,  t  he  re  fore,  in  order  to 

obtain  the  complete  value  of  N,  we  must  subtract  some  positive  number  from 
the  sum  a — b+c — d+e—f+g.     Hence 

a—b+c— d+e—f+g>N, 
and  it  has  been  shown  that 

\  a  —  b  +  c—d+e—f        <N; 

therefore  N  lies  between  these  two  sums. 

From  this  it  follows  that,  since  g  is  the  numerical  value  of  the  difference 
of  these  two  sums,  the  error  committed  ivhcn  we  assume  a  certain  number  of 
terms  a — b-\-c — d-\-e—f  for  the  value  of  N  is  numerically  less  than  the  term 
which  immediately  follows  that  at  which  we  stopped. 

In  the  preceding  example,  all  the  terms  after  the  first  being  alternately  posi- 
tive and  negative,  we  may  conclude  that  the  numerical  value  of  the  first  five 
terms 

4         16  320  \'">60 

3  +  27_  2187  "^"531441  ~43046721 
differs  from  tho  true  value  of  V^l   uy  a  quantity  less  than  the  value  of  the 

112640 

sixth  term,  which  was  found  to  bo  equal  to ;  but  this  fraction  is 

17433922005 

by   mere    inspection   less   than  ,   therefore,   when   we    assume    that 

V31=3. 14138,  the  result  is  within  0.00001  of  the  truth. 

117.  From  what  has  been  said  above  it  will  be  seen  that,  in  order  to  obtain 
an  approximate  value  of  tho  n"'  root  of  any  number  N  by  the  method  of  series, 
we  may  make  use  of  the  following 

rum:. 

Resolve  the  given  number  N  into  two  parts  of  the  form  pn  -f-  q.  whrrc  p"  is  the 

i 
highest  u"'  power  contained   in  N,  and  in   t)ir  development  of  (x-f-a)"  make 


DEGREE  OF  APPROXIMATION  OF  SERIES.  117 

x=pn,  a=q.     The  number  of  terms  to  be  taken  in  the  resulting  series  wil. 

depend  upon  the  degree  of  accuracy  required,  and  can  be  determined  by  the 

principle  just  explained.     Convert  all  the  terms  of  which  account  is  taken  into 

decimals,  and  then  effect  the  reduction  between  the  additive  and  subtractive 

terms. 

q 
This  method  can  not  be  employed  with  advantage  except  when  —  is  a  small 

fraction ;  for  unless  this  be  the  case,  the  terms  of  the  series  will  not  diminish 

with  sufficient  rapidity,  and  it  will  be  necessary  to  take  account  of  a  great 

number  of  terms  in  order  to  arrive  at  a  near  approximation. 

It  may  happen  thatp"  is  <^q ;  we  must  then  modify  the  above  process,  for 

pa       a  a      .    . 

then  —  or  —  is  greater  than  unity,  and  therefore  all  the  powers  of  -  will  m 

crease  in  numerical  value  as  the  degree  of  the  power  increases. 

Suppose,  for  example,  that  the  cube  root  of  56  is  sought,  27  being  the 
greatest  cube  contained  in  56,  wo  shall  have 

a     29 

x=27,  a  =  29  and  .-.  -=-, 

nnd  the  terms  of  the  series  will  go  on  increasing  instead  of  diminishing  (we  do 
not  speak  of  the  coefficients,  which  are  fractions  differing  but  little  from  unity). 

8         1  „       . 

But  we  may  resolve  56  into  64 — 8,  or  43 — 8  ;  but  — ,  or  -,  is  a  small  fraction. 

b4        8 

On  the  other  hand,  if  we  substitute  — a  for  a  in  the  expression  for  Vr+a> 

we  have 

r         1    a     1    Ti  —  1    a2     1    n— 1    In— 1    a3 


8/ '  x  —  6£:=£n(l . . 

n    x     n      2n      x'2     n      2n         3n       x3 
If  we  put  x=64,  a=8,  we  shall  obtain  a  series  of  terms  which  will  de- 
crease with  great  rapidity. 

Here  all  the  terms,  with  the  exception  of  the  first,  are  negative,  and  we  can 
not  apply  to  this  scries  the  criterion  established  in  (Art.  116)  for  fixing  the  de- 
gree of  approximation.  But  we  shall  approach  very  nearly  to  the  required 
degree  of  approximation  if  we  take  into  account  such  a  number  of  terms  that 
the  first  which  we  neglect  shall  be  less,  by  one  tenth,  for  example,  than  the 
decimal  place  to  which  we  wish  to  limit  the  approximation. 
The  student  may  take  the  following  examples  as  exercises : 

(1)  V39   =V32  +7   =2.0607 true  to  0.0001. 

(2)  3/65   =  V64   +1   =4.02073  . . .  true  to  0.00001. 

(3)  {/260=  V 256  +  4   =4.01553  ...  true  to  0.00001. 

(4)  yil)8  =  V128— 20=1.95204  .  . .  true  to  0.00001. 

118.  Roots  of  imaginary  expressions  of  the  form  a±6  -J  —  1  are  extracted 


by  putting  the  expression  under  the  form  (a  ±6  V  —  l)n,  and  developing  by  the 
binomial  theorem ;  a  series  of  terms  will  thus  be  obtained,  which  may  be  put 
under  the  form  A+B  yf  —  1,  A  representing  the  algebraic  sum  of  the  rational 
terms,  and  B  the  algebraic  sum  of  the  coefficients  of  V  —  1-  Algebra  fur- 
nishes no  other  general  method  for  this  transformation,  but  when  n  is  a  power 
of  2,  it  can  be  effected  without  the  aid  of  series 


lid  ALGEBRA. 


Let  us  consider,  first,  the  two  radicals  yJa-\-b  y/  —  1  and  \!  a  —  by/    -1. 
Placing 

[1]  y]a+by/^-L+ya  —  by/~]=.r 

[2]  y]a+byf— I— Va— 6/3i=y, 

and  squaring  both,  there  results 

2a+2-/a-!-H2=xs 
2a— 2y/a^fb1=y2. 

Whatever  may  be  the  sign  of  a,  the  value  of#x:  is  positive,  but  that  of  y*  is 
negative.     From  these  equalities  we  derive 

[3]  .T=V2a+2  y/at+V,  y=\y] —Qa+2y/a*+bV  yf^l. 
But  the  equalities  [11  and  [2]  givo 

Then,  finally,  putting  for  a:  and  ?/  the  values  [3],  we  shall  have 
[4]  Ja+by/~l=     lj2a  +  2y/^f& 

+  ±J-2a  +  2  Va2+i2  V^ 

[5]   Ja-b  y/~=l=      ^2a  +  2V^4^ 

--l-2a  +  2y/a'1+b"'  /^T. 
Now,  if  we  consider  the  radical  expressions 

\a±b^~l,y]a±:by/~—i,  y  a±b  y/~—i,  &c, 
we  observe  that  the  extraction  of  a  single  root  which  is  some  power  of  two, 
can  be  replaced  by  successive  extractions  of  the  square  root;  consequently, 
the  repetition  of  the  formulas  [4]  and  [5]  will  always  reduce  the  above  ex- 
pressions to  expressions  of  the  form  AJLB  y/ — 1. 

Remark. — In  each  of  these  formulas  tho  first  member,  by  reason  of  th 
radicals  which  it  contains,  may  have  four  different  values,  and  the  same 
true  of  the  second  member.  In  both,  tho  lour  values  of  the  first  member  arc 
the  same,  and  this  is  tho  case  evidently  with  tho  second  member;  so  that 
tho  two  formulas  make  really  but  one.  They  present  no  difference  except 
when  wo  use  them  simultaneously  in  the  same  algebraical  calculation,  because 
then  we  ought  to  regard  tho  terms  into  which  y/ — 1  enters  us  affected  with 
contrary  signs.  But  then  it  is  necessary  to  remark  besides,  that,  by  tho  very 
manner  in  which  wo  have  arrived  at  these  formulas,   yf  a--\-l>:  in  them  re  pre 

sents  the  product  of  yJa-{-b  y/  —  1  \!a  —  b  yf —  I  ;  consequently,  the  del 
inutions  of  these  two  radicals  ought  always  t<>  be  snppo      1  I    OCMfted  in  Ml 
a  manner  that  their  product  should  have  the  rigs  which  is  given  to  y/a:-\-u~ 
in  the  second  member.      Without  attention  to  this  tbo  formulas  might  lead  to 
false  results. 


RATIOS  AND  PROPORTION.  119 

Another  remark  of  importance  may  be  added  here. 

The  methods  of  proceeding  in  certain  operations  upon  imaginary  exiressions, 
exhibited  at  (Art.  6G),  were  suited  to  the  restrictions  which  in  ordinary  cases 
would  be  understood  as  pertaining  to  the  radical  sign.  If,  however,  this  sign 
have  its  most  general  signification,  it  must  be  used  in  its  ambiguous  sense, 
that  is,  as  having  JL  before  it.  Then  \/ — flX  V — a  would  have  a  more  ex- 
tended sense  than  simply  the  square  of  ■/  — a.  It  would  have,  in  fact,  four 
values, 
-j_  y/  —a  x  +  V  —  a,    —  V  —a  x  +  V  —  a,    +  V  —  «X-  V  ~  a> 

—  V  —  ax  —  V— «> 

or 

—a,  +c,  +a,  —a 

These  four,  in  fact,  amount  to  but  two,  -\-a  and  — a,  which  are  the  values 
obtained  by  the  ordinary  rale  of  multiplication,  -/ — ax  V — a=  •v/a2=ia- 

If  the  quantities  under  the  radical  are  different,  the  reasoning  will  be  a  little 
varied.     Let  the  product  be  required  of 


The  first  of  these  factors  •/  — a  may  be  put  under  the  form  a'  V  — 1,  and 
the  second  under  the  form  b'  V  —  1.     The  product  will  then  be  expressed  by 

a'b'  t/~— 1 X  V— 1- 
But  after  what  has  just  been  said,  if  there  be  no  restriction  in  the  meaning 
of  the  sign  -/     ,  we  have  -/ — lxV — l  =  rtl.     Hence 

a'6'/^lX  V~^-L  =  ±a'b'. 
But  since  the  square  of  a'b'  is  a'-b'2,  or  ab,  we  have  a'b'=  y/ab,  and,  there 

fore,  

V — aX  V — 6=i  Vab, 
the  result  winch  we  should  obtain  by  the  ordinary  rule  for  the  multiplies 
tion  of  radicals.     We  thus  perceive  that  this  rule  gives  us  the  true  product 
in  its  most  general  form  wh,en  there  is  no  restriction  in  the  sense  of  the  radi- 
cal sign. 


RATIOS  AND  PROPORTION. 


119.  Numbers  may  be  compared  in  two  ways. 

When  it  is  required  to  determine  by  how  much  one  number  is  greater  or 
less  than  another,  the  answer  to  this  question  consists  in  stating  the  difference 
between  these  two  numbers.  This  difference  is  called  the  Arithmetical  Ratio 
of  the  two  numbers.  Thus,  the  arithmetical  ratio  of  9  to  7  is  9— 7,  or  2,  and 
if  a,  b  designate  two  numbers,  their  arithmetical  ratio  is  represented  by  a — b. 

When  it  is  required  to  determine  how  many  times  one  number  contains,  of 
is  contained  in  another,  the  answer  to  this  question  consists  in  stating  the 
quotient  which  arises  from  dividing  one  of  these  numbers  by  the  other.  This 
quotient  is  called  the  Geometrical  Ratio  of  the  two  numbers.  The  term 
Ratio,  when  used  without  any  qualification,  is  always  understood  to  signify  a 
geometrical  ratio,  and  we  shall,  at  present,  confine  our  attention  to  ratios  of 
this  description. 


120  ALGEBRA. 

120.  By  the  ratio  of  two  numbers,  then,  we  mean  the  quotient  which  arises 

from  dividing  one  of  these  numbers  by  the  other.     Thus,  the  ratio  of  12  to  4 

12  5  1 

is  represented  by  —  or  3,  the  ratio  of  5  to  2  is  -  or  2.5,  the  ratio  of  1  to  3  is  - 

or  .333 . . .  We  here  perceive  that  the  value  of  a  ratio  can  not  always  be  ex- 
pressed exactly,  except  in  the  form  of  a  vulgar  fraction,  but  that,  by  taking  a 
sufficient  number  of  terms  of  the  decimal,  we  can  approach  as  nearly  as  we 
please  to  the  true  value. 

121.  If  a,  b  designate  two  numbers,  the  ratio  of  a  to  b  is  the  quotient 

arising  from  dividing  a  by  b,  and  will  be  represented  by  writing  them  a  :  b,  or  r. 

122.  A  ratio  being  thus  expressed,  the  first  term,  or  a,  is  called  the  ante- 
cedent of  the  ratio;  the  last  term,  or  b,  is  called  the  consequent  of  the  ratio. 

123.  It  appears,  therefore,  that,  iu  arithmetic  and  algebra,  the  theory  of 
ratios  becomes  identified  with  the  theory  of  fractions,  and  a  ratio  may  be  de- 
fined as  a  fraction  whose  numerator  is  the  antecedent,  and  whose  denominator 
is  the  consequent  of  the  ratio. 

124.  When  the  antecedent  of  a  ratio  is  greater  than  the  consequent,  the 
ratio  is  called  a  ratio  of  greater  inequality ;  when  the  antecedent  is  less  than 
the  consequent,  it  is  called  a  ratio  of  less  inequality ;  and  when  the  antecedent 

12 

and  consequent  are  equal,  it  is  called  a  ratio  of  equality.     Thus,  —  is  a  ratio 

12  3 

of  greater  inequality,  —rj  is  a  ratio  of  less  inequality,  -  or  1  is  a  ratio  of 

equality.  It  is  manifest  that  a  ratio  of  equality  may  always  be  represented  by 
unity. 

125.  When  the  antecedents  of  two  or  more  ratios  are  multiplied  together 
to  form  a  new  antecedent,  and  their  consequents  multiplied  together  to  form 
a  new  consequent,  the  several  ratios  are  said  to  be  compounded,  and  the  re- 
sulting ratio  is  called  the  sum  of  the  compounding  ratios.     Thus,  the  ratio  t 

c 
is  compounded  with  the  ratio  -,  by  multiplying  the  antecedents  a,  c  for  a  new 

antecedent,  and  the  consequents  b,  d  for  a  new  consequent,  and  the  resulting 

ac  a         c 

ratio  t-?  is  called  the  sum  of  the  ratios  r  and  -.. 
bd  b  d 

m  p  r    t 
In  like  manner,   the   ratios  -,  -,  -,  —  are  compounded  by  multiplying  all 

the  antecedents  together  for  a  new  antecedent,  and  all  the  consequents  for  a 

mprt 
new  consequent,  and  the  resulting  ratio,  — — ,  is  called  the  sum  of  the  ratios 
1  °  nqsw 

mprt 

n'  7'  s'  w' 

12G.  When  a  ratio  is  compounded  with  itself  tho  resulting  ratio  is  called  the 

d indicate  ratio,  or  double  ratio  of  the  primitive.     Thus,  if  we  compound  the 

a  a  fl'  i  a 

ratio  j  with  j,  tho  resulting  ratio,  vs,  is  called  tho  duplicate  ratio  of  -r- 

a3  a 

Similarly  tj  is  called  tho  triplicate  ratio,  or  triple  ratio  of  r- 


RATIOS  AND  PROPORTION.  121 

an  .      a 

And,  generally,  j~  is  called  the  sum  of  the  n  ratios  y- 

1 

a2 
\ccording  to  the  same  principle,  the  ratio  —  is  called  the  subduplicale  ratio, 

b2 

X  1        1 

fa  a2  .    a2     a2     a 

-■'  half  ratio  of  y-;  for  the  duplicate  ratio  of—  is  —  X- f=r» 

b  b2      b2     b2 

i 
a3 
So,  also,  the  ratio  — j  is  called  the  sublriplicate  ratio,  or  one  third  of  the  ratio 

b? 

I  I  J.  A 

ea      -r,  .  ,         .      na3      a*     aJ     a3     a 

ot  r«     r  or  the  triple  ratio  of  —  is  —  X  —  X  "T=T* 

b3      Vs     h*     b3 

i 

a"  a 

And,  in  general,  —  is  called  one  nth  of  the  ratio  r ;  for  n  times  the  ratio 

b° 
liii  , 

aa .    an     an     a»  a 

I's-X-X-X--  ton  terms  =y-. 

ba      b"      o"     6" 

3 

Note. — The  ratio  — g  is  called  the  sesquiplicate  ratio  of  y-,  for  it  is  com- 

I  1 

a2     a     a3 
psunded  of  the  simple  and  subduplicato  ratio  ;  thus,  —  Xr=— • 

b2     b      b2 

127.  If  the  terms  of  a  ratio  be  both  multiplied,  or  both  divided,  by  the  same 

quantity,  the  value  of  the  ratio  remains  unchanged. 

a 
The  ratio  of  a  to  6  is  represented  by  the  fraction  7  ;  and  since  the  value  of 

a  fraction  is  not  changed,  if  we  multiply,  or  divide,  both  numerator  and  de- 
nominatoi  by  the  same  quantity,  the  truth  of  the  proposition  is  evident.    Thus, 

a 

a_ma__n  or  a:o=roa:m&=-:-. 

b~mb~b  n   n 

n 

128.  Ratios  are  compared  with  each  other  by  reducing  the  fractions,  by 
which  they  are  represented,  to  a  common  denominator. 

[f  wo  wish  to  ascertain  whether  the  ratio  of  2  to  7  is  greater  or  less  than 

2  3 

that  of  3  to  8,  since  these  ratios  are  represented  by  the  fractions  -  and  -, 

/  8 

which  are  equivalent  to  —  and  —  ;  and  since  the  latter  of  these  is  greater  than 

the  former,  it  appears  that  the  ratio  of  2  to  7  is  less  than  the  ratio  of  3  to  8. 

129.  A  ratio  of  greater  inequality  is  diminished,  and  a  ratio  of  a  less  inequal- 
ity is  increased,  by  adding  the  same  quantity  to  both  terms. 


122  ALGEBRA. 

Let  j  represent  nny  ratio,  and  let  x  be  added  to  each  of  its  terms.     The 

two  ratios  will  then  bo 

a  a-\-x 

V  b+x' 

which,  reduced  to  a  common  denominator,  become 

ab-\-ax    ab-ifbx 

b(b+xy  b{b+x)' 

a 

If  a>6,  i.  e.,  if  t  be  a  ratio  of  greater  inequality,  then 

ab-\-ax      ab-\-bx 
b{b+xyb{b+x); 

md  ••.  r  is  diminished  by  the  addition  of  the  same  quantity  to  each  of  its  term*. 

a 

Again,  if  a<b,  i.  c,  if  t  be  a  ratio  of  less  inequality,  then 

ab-\-ax      ab-\-bx 
b(b+x)<b{b+x)' 

and  ••.  j  is  increased  by  the  addition  of  the  same  quantity  to  each  of  its  terms. 

130.  If  there  be  any  number  of  ratios  in  which  the  consequent  of  the  first  ratio 
is  the  antecedent  of  the  second,  and  the  consequent  of  the  second  the  antecedent 
of  the  third,  and  so  on,  the  sum  of  any  number  of  said  ratios  is  the  ratio  of  the 
first  antecedent  to  the  last  consequent. 
Let  the  proposed  ratios  be 

a    b    c   d  e 
F"?  3' ?/•"""• 
Then,  by  (Art.  125),  their  sum  is 

a     b      c     d     e 

bx!xdxlxf"~' 
or 

abede 


bedef----' 

a 
i.  e.,j. 


131.  Proportion  is  an  equality  of  ratios. 

Thus,  if  a,  b,  c,  d  be  four  quantities,  such  that  a,  when  divided  by  b,  gives  the 
same  quotient  as  c  when  divided  by  d,  then  a,  b,  c,  d  are  said  to  be  in  propor- 
tion, or  to  bo  proportionals ;  the  numbers  20,  5,  36,  9  are  proportionals,  for 

20  .  36 

-=4,and-=4. 

When  four  quantities  aro  proportionals,  it  is  usually  enunciated  by  saying 
that  the  first  is  to  the  second  as  the  third  is  to  the  fourth.  Thus,  if  a,  6,  c,  d  uro 
proportionals,  wo  say  that  a  is  to  b  as  c  is  to  d,  and  this  is  expressed  lyr  wri- 
ting them 

a:b::c:d,  or  a:b=c:d, 
or  as  fractions, 

a       c 
~b~d' 


RATIOS  AND  PROPORTION.  123 

The  first  or  second  form  of  notation  is  usually  employed  in  geometry,  the 
last  in  analytical  investigations.  The  signs  : :  and  =  have  precisely  the  same 
meaning.     The  sign  :  is  the  sign  of  division. 

a      c 

132.  The  expression  a :  b : :  c :  d,  or  7  =-7,  is  called  a  proportion,  and  a,  b,  c,  d 

are  severally  called  the  terms  of  the  proportion.  The  first  and  last  are  called 
the  extreme  terms,  the  second  and  third  the  mean  terms.  The  first  term  is 
called  the  first  antecedent,  the  second  term  t\\a  first  consequent,  the  third  terra 
the  second  antecedent,  and  the  fourth  term  the  second  consequent. 

133.  When  the  second  and  third  terms  of  a  proportion  are  identical,  the 
quantity  which  forms  these  terms  is  called  a  mean  proportional  between  the 
other  two ;  thus,  if  we  have  three  quantities  a,  b,  c,  such  that 

a     1  a      b 

a:b::b:c,  or  r==-, 

b      c 
then  b  is  said  to  be  a  mean  proportional  to  a  and  c,  and  c  is  called  a  third  pro 
porlional  to  a  and  b. 

If,  in  a  series  of  proportional  magnitudes,  each  consequent  be  identical  with 
the  next  antecedent,  these  quantities  are  said  to  be  in  continued  proportion ; 
thus,  if  we  have  a  series  of  quantities,  a,  b,  c,  d,  e,f,  g,  h,  such  that 

a :  b : :  b :  c : :  c :  d : :  d :  e : :  e  :f:  :f:  g : :  g :  h , 
or 

a     b      c     d     e     f     g 
b     c     d      e     /     g     A' 
then  the  quantities  a,  b,  c,  d,  e,f,  g,  h  are  in  continued  proportion. 
A  continued  pi'oportion  is  called  a  progression. 

The  following  are  the  most  important  propositions  connected  with  the  sub- 
ject of  proportion. 

I.  If  four  quantities  be  proportionals,  tJie  product  of  the  extreme  terms  vnll  be 
equal  to  the  product  of  the  mean  terms. 

Let 

a:b::c:d, 

or 

a_c 

b~d' 

Multiplying  these  equals  by  bd,  the  expression  becomes 

ad=bc. 

II.  Conversely,  If  the  product  of  any  two  quantities  be  equal  to  the  product 
oj  any  other  two,  these  four  quantities  will  constitute  a  proportion,  the  terms  of 
one  of  Hie  products  being  the  means,  and  the  terms  of  the  other  the  extremes. 

Let 

ad=zbc. 
Dividing  these  equals  by  bd,  the  expression  becomes 

a     c         c      a 
b=d,0rd==b; 
i.  e.,  a :  b : :  c :  d,  or  c :  d : :  a :  b. 
In  the  first,  a  and  0  are  the  extremes,  and  b  and  c  the  means ;  in  the  second, 
b  and  c  are  the  extremes,  and  a  and  d  the  means. 

III.  If  three  quantities  be  in  continued  proportion,  the  product  of  the  extreme 
tcrn\s  is  equal  to  the  square  of  the  mean. 


124  ALGEBRA. 

This  follows  immediately  from  I. ;  for  let  a,  b,  c  be  three  quantities  in  con- 
tinued proportion,  then 

7        7  a       b 

a:b::b:c,  or  v=- 
b      c 

.-.  acz=b  X  b  by  I. 

=  b*,  or  b=  y/ac. 

IV.  Conversely,  If  the  product  of  any  two  quantities  be  equal  to  the  square 
of  a  third,  the  last  quantity  will  be  a  mean  proportional  between  the  otfier  two 

Thus,  if  ac=b-,  b  is  a  mean  proportional  between  a  and  c  ;  for,  since 

ac=b2, 
dividing  these  equals  by  be, 

-=-,  or  a:b::b:c. 
b      c 

V.  Quantities  which  have  the  same  ratio  to  the  same  quantity  are  equal  to 
one  another,  and  those  to  which  the  same  quantity  has  the  same  ratio  are  equal 
to  one  another. 

First,  let  a  and  b  have  the  same  ratio  to  the  same  quantity  c,  then  a  =  b. 
Since 

a:c::b:c, 
or 

a     b 
c=c; 
multiply  these  equals  by  c  .•.  a=b. 

Again,  let  c  have  the  same  ratio  to  each  of  the  quantities  a  and  b,  then  a=J> 
Since 

c:a::c:b, 
or 

c     c 
a~V 
dividing  these  equals  by  c, 

11 
a     b 
.-.  a=b. 

VI.  Ratios  that  are  equal  to  the  same  are  equal  to  one  another. 
Let  a:b::x:y 


c  Then  a:b::c:d. 
And  c:d::x:y 

This  is  an  axiom. 

VII.  If  four  quantities  be  j>roportionals,  they  will  be  proportionals  edso  alter 
nando,  that  is,  the  first  will  have  the  same  ratio  to  the  third  that  the  setend  hat 
to  the  fourth. 

Let  a:b::c:d,  then,  also,  a:c::b:d. 

a     c 
Since  t=;7.  divide  each  of  these  equals  by  c,  and  multiply  each  by  b 

Then  -=-j»"  i.  e.,  a:c::b:</. 

c      d 

VIIT.   If  four  quantities  be  proportionals,  they  will  be  proportionals  also 

invertrmlo,  that  is,  the  second  ivill  have  to  tin-  first  the  same  ratio  that  the 

fuuitli  hat  to  thf  third. 


RATIOS  AND  PItOPOilTION.  X25 

Let  a:b::c:d,  then,  also,  b:a::d:c. 

a     c 
Since  v=j   divide  unity  by  each  of  these  equals. 

We  have 

1  1 


©~Q' 


b    d 

— =— ;  1.  e.,  b:a::a:c. 
a     c 

IX.  If  four  quantities  be  proportionals,  they  will  be  proportionals  also  com- 
ponendo,  that  is,  the  first,  together  with  the  second,  will  have  to  the  second  the 
same  ratio  that  the  third,  together  with  the  fourth,  has  to  the  fourth. 

Let  a:b::c:d,  then,  also,  a-\-b:b::c-\- d:d. 


or 


Since  T=7'  ac^  1  to  eaca  °f  these  equals,  then 

a  c 

b  +  ^d 

a-\-b      c-\-d 


a  c 

1  +  1=^+1, 


d 


;  i.  e.,  a-\-b:b::c-\-d:d. 


X.  If  four  quantities  be  proportionals,  they  will  be  'proportionals  also  divi- 
dendo,  that  is,  the  difference  of  the  first  and  second  will  have  to  the  second  the 
same  ratio  that  the  difference  of  the  third  and  fourth  has  to  the  fourth. 

Let  a:b::c:d,  then,  also,  a — b:b::c — d:d. 

a     c 
Since        T—~r  subtract  unity  from  each  of  these  equals,  then 

a  c 

b-l=d~1' 

or 

a  —  b     c—d     . 

—t — = — j—  ;  l.  e.,  a  —  b:b::c — d:d. 
b  a 

XI.  If  four  quantities  be  proportionals,  they  will  be  proportionals  also  con- 
vertendo,  that  is,  the  first  will  have  to  the  difference  of  tSie  first  and  second  the 
same  ratio  that  the  third  has  to  the  difference  of  the  third  and  fourth. 

Let  a  :  o:  :c:d,  then,  also,  a:  a  —  b  :  :c:c — d. 

a     c  b      (I        ' 

Since  r= ji  then,  by  prop.  VIII.,  -=7  ;  aud  hence,  subtracting  iheso  equal 

Quantities  from  unity, 

b__        d 

a  c' 

or 

a  —  b      c  —  (I 
~a~  =  '~c~', 

* 

ar 

a  c 

r  = ; ;  i.  e..  a: a  —  b::c:c — d. 

a — 0     c  —  d 


126  ALGEBRA. 

XII.  If  four  quantities  be  proportionals,  the  sum  of  the  first  and  second  wiu 
have  to  their  difference  the  same  ratio  that  the  swn  of  the  third  and  fourth  hoi 
to  their  difference. 

Let  a:b::c:d,  then,  also,  a-\-b:a — b::c-\-d'.c — d. 

Since  T==~r  we  Uave» 

by  prop.  IX.,  —=-T; 

a  —  b     c — d 
and,  by  prop.  X.,  — -g- =— -j-; 

dividing  these  equals  by  each  other, 
a-\-b     c-\-d 


:-d 


or 


a-\-b     c-\-d 

7= -,;  i.  e.,  a-\-b:a — b::c4-d:c — d. 

a  —  o     c — d  '  ' 

XI I T.  If  there  be  any  number  of  quantities  more  than  two,  and  as  many 
others,  which,  taken  two  and  two  in  order,  are  proportionals  (ex  aequali),  the 
first  will  have  to  the  last  of  the  first  rank  the  same  ratio  Uiat  the  first  of  the 
second  rank  has  to  the  last. 


Let 
and 
Let 


a,  b,  c,  d  ....  be  any  number  of  quantities, 
e,f,g,h    ....  as  many  others. 


a:b  ::e  :f 

b:c  ::f:g{{ 

►  Then,  also,  a:d::e:h. 

c  :d::g:h. 

For,  since 

a     e 
b=f 

IJL 

c  g 

c      g 

d~h' 

multiplying  the  first  column  together,  and  also  the  second, 

abc      efg 

bcl^fglC 

<>r 

a     e 

77=r 

;  i.  e.,  a:d::e:h. 

\  1  V.  Jf  (here  be  any  number  of  quantities  more  than  two,  and  as  many 
nth'  r»,  which,  taken  tivo  and  tivo  in  a  cross  order,  arc  proportionals  (ox  tr-quali 
perturbata),  the  first  will  have  to  the  last  of  the  first  rank  the  same  tatio  that  the 
first  <>f  the  second  rank  has  to  the  last. 


RATIOS  AND  PROPORTION. 


127 


Let 
and 

Let 

For,  since 


a,  b,  c,  d  .  .  .  .  be  any  number  of  quantities, 
e,fg,h....&s  many  others. 


a :  b  ::g:h  i 

b : c  ::f  :g>  Then,  also,  a : d : : e : h. 

c  :d::e  :f  ) 


a 
bz 

g 
~h 

b 

c~ 

_f 

g 

c 

e 

d  = 

7' 

abc 
bed' 

or 

a     e     ■  i       i 

-j=T ;  i.  e.,  aid'.'.e'.h. 

XV.  If  four  quantities  be  proportionals,  any  powers  or  roots  of  these  quan 
tilies  will  also  be  proportionals. 

Let  a : b : : c : d ;  then,  also,  an :bn::ca: da. 

Since 

a     c  r.  ,  ,  /a\n      /c\" 

t=-j,  raising  each  of  these  equals  to  the  nth  power,  It)  =1 -7I, 

or 

an     cn 

^=jn;  i.  e.,  an:bn::ca:da, 

,    where  n  may  be  either  integral  or  fractional.* 

XVI.  If  there  be  any  number  of  proportional  quantities,  the  first  will  have  to 
the  second  the  same  ratio  that  the  sum  of  all  the  antecedents  has  to  the  sum  of 
all  the  consequents. 

Let  ",  b,  c,  d,  e,f,  g,  h  be  any  number  of  proportional  quantities,  such  that 

a :  b : :  c :  d : :  e  :f: :  g :  h. 


Then 
Since 

we  bave 


and 


or 


a :  b : :  a-{-c-±-e-{-g:  b-\-d-{-f-{-h. 

a     c      e      g 
l=d=f~h' 

ab  =ba 

ad=bc 

af=be 

ah  =  bg, 

a(b+d+f+h)  =  b\a+c+e+g) 

a     a-^-c-\-e-\-g 

'■  1—b  +  d+f+h 

a:b::a-\-e-\-e-\-g:b-\-d-\-f-\-h. 


•  The  ratio  of  the  resulting  proportion  is  tnc  »u  uower  o'  Jie  ratio  of  the  jjiven  proportion 


128  ALGEBRA. 

XVII.  If  three  quantities  be  in  continued  proportion,  the  first  wiU  have  to 
the  third  the  duplicate  ratio  of  that  which  it  has  to  the  second. 

Let  a :b::b:c,  then  a :c::a2: 6s. 

Since 

a     b  a 

j=~,  multiply  each  of  these  equals  by  j-;  then 

a     a     b     a        a-     a     . 
bXb=-cXb'or¥=c;  i-e.,a:c::a»:*». 

XVIII.  If  four  quantities  be  in  continued  proportion,  the  first  will  have  to 
the  fourth  the  triplicate  ratio  of  that  which  it  has  to  the  second. 

Let  a,  b,  c,  d  be  four  quantities  in  continued  proportion,  so  that 

a:b::b:c::c:d  ;  then,  also,  a : d : : a3 : b3. 
Since 

a     b     c 


b~c—  d,V> 

e  i 

lave 

» 

a     b 

b=c 

a     c 

b~d 

a     a 

b=b' 

Multiplying 

these 

equals 

together, 
a3     bca 
bi=7d~V 

or 

a3     a 
l~3=d  '  U  6" 

,  a 

:d: 

:a3 

:6s, 

XIX.  If  two  proportions  be  multiplied  together,  term  by  term,  the  products 
will  form  a  proportion. 

Let  a:  b  ::  c  :d, 

and  e:f::g:h; 

then  ae:bf::cg:dh, 

a     c  c       g 

for  T  =  ~n  a,u'  7=rj 


hence,  multiplying  equals, 


b-d'a"\f 


ae      eg 

—=—  or  ae:bf:iCg:dh.* 

Tho  compatibility  of  any  change  in  the  on|er  of  the  terms  of  a  proportion 
may  be  tested  by  forming  the  product  of  the  extremes  and  means  in  both  the 
original  and  changed  proportion,  when,  if  they  Agree,  the  change  is  correct 
Thus.  a:b::c:d  may  be  written  d:b::c:>i,  for  we  hare  ad=zbc  in  both. 

I  \  IMPLES   in    PROPORTION. 

(1)  The  mercurial  barometer  stands  a)  a  height  of  30  inches,  and  the  specific 
gravity  of  quicksilver  is  ].'?'•, ;!.      How  high  would  a  water  barometer  stand  .' 

\xa.  ."•".  feel  1 1 1  inches. 

i  i  The  weights  Of  a  lever  have  the  same  ratio  as  the  lengths  of  the  oppo 
site  arms.  The  ratio  of  the  weights  is  .">,  and  the  longer  arm  10  indies 
What  is  tin-  length  of  the  Bhorter  arm  .'  a\ns.  2  inches. 

*  The  ratio  of  the  resulting  proportion  la  ti  I  1 1 •  ■  -  ratios  of  the  two   fivi  d  i    , 

portions. 


EQUATIONS  129 

(3)  The  weights  of  a  lever  are  6  and  8  pounds,  and  the  length  of  the  .shorter 
arm  18  inches.     What  is  that  of  the  longer  ?  Ans.  24  inches. 

(4)  At  the  end  of  an  arm  of  a  lever  5  inches  long,  what  weight  can  be  sup- 
ported by  2\  pounds  acting  at  the  end  of  an  arm  4|  inches  long? 

Ans.  2485  pounds. 

(5)  Triangles  are  to  each  other  as  the  products  of  their  bases  by  their  alti- 
tudes. The  bases  of  two  triangles  are  to  each  other  as  17  and  18,  and  their 
altitudes  as  21  and  23.     What  is  the  ratio  of  the  triangles  themselves  1 

Ans.  119:138. 

(6)  The  force  of  gravitation  is  inversely  as  the  square  of  the  distance.  At 
the  distance  1  from  the  centre  of  tho  earth  this  force  is  expressed  by  the 
number  32.16.     By  what  is  it  expressed  at  the  distance  60  ? 

Ans.  0.0089. 

(7)  Tho  motion  of  a  planet  about  the  sun  for  a  short  space  is  proportional 
to  unity  divided  by  the  duplicate  of  the  distance.  If  the  motion  be  represented 
by  v  when  the  distance  is  r,  by  what  will  it  be  expressed  when  the  distance  is  r1 1 


Ans.— 


'j 


(8)  The  times  of  revolution  of  the  planets  about  the  sun  are  in  the  sesquipli- 
cate  ratio  of  their  mean  distances.  The  mean  distance  of  the  earth  from  the 
sun  being  expressed  by  1,  that  of  Jupiter  will  be  5.202776 ;  the  time  of  revolu- 
tion of  the  earth  is  365.2563835  days.  What  is  the  time  of  revolution  of 
Jupiter  ?  Ans.  4332.5848212  days. 


EQUATIONS. 

PRELIMINARY    REMARKS. 

134.  An  equation,  in  the  most  general  acceptation  of  the  term,  is  composed 
of  two  algebraic  expressions  which  are  equal  to  each  other,  connected  by  the 
sign  of  equality. 

Thus,  ax=b, cx*-\-dx=e,  cx3-{-gx':=hx^-k,  mxi+nx3-{-px'i+qx-\-r=0,aTe 

equations. 

The  two  quantities  separated  by  the  sign  =  are  called  the  members  of  the 
equation,  the  quantity  to  the  left  of  the  sign  =  is  called  the  first  member,  the 
quantity  to  the  right  the  second  member.  The  quantities  separated  by  the 
signs  4-  and  —  are  called  the  terms  of  the  equation. 

135.  Equations  are  usually  composed  of  certain  quantities  which  are  known 
and  given,  and  others  which  are  unknown.  The  known  quantities  are  in 
general  represented  either  by  numbers,  or  by  the  first  letters  in  the  alphabet, 
a.  b,  c,  &c. ;  the  unknown  quantities  by  the  last  letters,  s,  t,  x,  y,  z,  &c. 

136.  Equations  are  of  different  kinds. 

1°.  An  equation  may  be  such  that  one  of  the  members  is  a  repetition  of  the 
other;  as,  2x— 5=2r— 5. 

2°.  One  member  may  be  merely  the  result  of  certain  operations  indicated 
in  tho  other  member;  as,  5a:-r-16=10x— 5  —  (5a:— 21),  (x+y){x— ?/)=x3— y\ 
r5 — i/3 

— 2-=X*  +  Xy  +  y* 


ALGEBRA. 

.  All  the  quantities  in  each  member  may  be  known  and  given;  as,  2.3  =  10 
+  15,  a-{-b=c— d,  in  which,  il"  we  substitute  fur  a,  b,  c,  d  tho  known  quan- 
tities which  they  Represent,  the  equality  subsisting  between  the  two  members 

will  bo  sell-evident. 

In  each  of  the  above  cases  the  equation  is  called  ntical  equation. 

4  .  Finally,  the  equation  may  contain  both  known  and  unknown  quantities, 
and  bo  such  that  the  eufhality  subsisting  between  the  two  members  can  not  be 
made  manifest,  until  we  substitute  for  the  unknown  quantity  or  quantities  cer- 
tain other  numbers,  the  value  of  which  depends  upon  the  known  numbers 
which  enter  into  the  equation.  The  discovery  of  these  unknown  numbers 
constitutes  what  is  called  tho  solution  of  the  equation. 

When  found  and  put  in  the  place  of  the  letters  which  represent  them, 
if  they  make  the  equality  of  the  two  members  evident,  the  equation  is  said  to 
be  verified,  or  satisfied. 

The  word  equation,  when  used  without  any  qualification,  is  always  under- 
stood to  signify  an  equation  of  this  last  species  ;  and  these  alone  are  the  objects 
of  our  present  investigations. 

ar-|-4=:7  is  an  equation  properly  so  called,  for  it  contains  an  unknown 
quantity  x,  combined  with  other  quantities  which  are  known  and  given,  and 
the  equality  subsisting  between  the  two  members  of  tho  equation  can  not  be 
made  manifest  until  we  find  a  value  for  x,  such  that,  when  added  to  4,  the 
result  will  be  equal  to  7.  This  condition  will  be  satisfied  if  we  make  x=3 ; 
and  this  value  of  x  being  determined,  the  equation  is  solved. 

The  value  of  tho  unknown  quantity  thus  discovered  is  called  the  root  of  the 
equation,  being  the  radix  out  of  which  the  equal 'um  is  formed;  the  term  root 
hero  has  a  different  sense  from  thai  in  which  we  have  hitherto  used  it,  viz., 
that  of  the  base  of  a  power. 

137.  Equations  are  divided  into  degrees  according  to  the  highest  power  of 
the  unknown  quantity  which  they  contain.  Those  which  involve  the  simple 
or  first  power  only  of  tho  unknown  quantity  are  called  simple  equations,  or 
equations  of  the  first  degree;  those  into  which  the  square  of  the  unknown 
quantity  enters  are  called  quadratic  equations,  or  equations  of  the  second  de- 
gree: so  wo  have  cubic  equations,  or  equations  of  (he  third  degree  ;  biquad- 
ratic equations,  or  equations  of  the  fourth  degrt  t  ;  equations  of  the  fifth,  sixth, 

....  7ith  degree.     Thus, 

ax -\-b    =cx-\-d  is  a  simple  equation. 

4i* — 2x  =5 — a:2  is  a  quadratic  equation. 

x3-\-px2=z'2q  is  a  cubic  equation. 

xn-{-j)xn~1-\-qxn~2-{-,  &c,  =r,  is  an  equation  of  the  n"'  degree. 

138.  Numerical  equations  are  those  which  contain  numbers  only,  in  addition 
to  tin-  unknown  quantities.  Thus,  .r  -f  ,r)./-  =  r..r-f-17  anil  ■lx=7y  are  numer- 
ical equations. 

I.  Lateral  equations  are  those  in  which  tin-  known  quantities  are  repre 
sented  by  letters  only,  or  by  both  letters  and  cumbers.  Thus,  x  -\-j>.i:-\-<i.r=.r 
r* — Xj)xfy-\-!jqx^if:-\-rxfz=z!')  are  literal  equations. 

140.  Lot  us  now  pass  on  to  consider  the  solution  of  equations,  it  being  under- 
stood that  /"  sol/oe  an  <  quation  is  t<>  find  the  oalui  of  the  unknown  quantity,  or 
to  find  a  number  which,  when  substituted  for  the  unknown  <juantity  in  the  equa- 
tion, renders  On  first  member  identical  u-itii  the  second. 


SIMPLE  EQUATIONS.  131 

The  difficulty  of  solving  equations  depends  upon  the  degree  of  the  equations 
and  the  number  of  unknown  quantities.  We  first  consider  the  most  simple 
case. 

ON  THE  SOLUTION  OP  SIMPLE  EQUATIONS  CONTAINING  ONE  UN- 
KNOWN QUANTITY. 

141.  The  various  operations  which  we  perform  upon  equations,  in  order  to 
arrive  at  the  value  of  the  unknown  quantities,  are  founded  upon  the  following 
axioms : 

//'  to  two  equal  quantities  the  same  quantity  be  added,  the  sums  will  be  equal. 

If  from  two  equal  quantities  the  same  quantity  be  subtracted,  the  remainders 
will  be  equal. 

If  two  equal  quantities  be  multiplied  by  the  same  quantity,  the  products  will 
be  equal. 

If  two  equal  quantities  be  divided  by  the  same  quantity,  the  quotients  will  be 
equal. 

These  axioms,  when  applied  to  the  two  equal  quantities  which  constitute 
the  two  members  of  every  equation,  will  enable  us  to  deduce  from  them  new 
equations,  which  are  all  satisfied  by  the  same  value  of  the  unknown  quantity, 
and  which  will  lead  us  to  discover  that  value. 

142.  The  unknown  quantity  may  bo  combined  with  the  known  quantities  in 
the  given  equation  by  the  operations  of  addition,  subtraction,  multiplication 
and  division.     We  shall  consider  these  different  cases  in  succession. 

T.  Let  it  be  required  to  solve  the  equation 

x-\-a=b. 
If,  from  the  two  equal  quantities  x-\-a  and  b,  we  subtract  the  same  quantity 
a,  the  remainders  will  be  equal,  and  we  shall  have 

x-\-a — a=b — a, 
or 

x=b — a,  the  value  of  x  required. 

So,  also,  in  the  equation 

x+6=24. 

Subtracting  6  from  each  of  the  equal  quantities  x-\-  6  and  24,  the  result  is 

x=24  — G 
:=18,  the  value  of  x  required. 

II.  Let  the  equation  be 

x — a=b. 
If,  to  the  two  equal  quantities  .r — a  and  b,  the  same  quantity  a  be  aided, 
the  sums  will  be  equal ;  then  we  have 

x — a-{-a=b-\-a, 
or 

x=b-\-a,  the  value  of  x  required. 

So,  also,  in  the  equation 

a:— 6=24. 
Adding  6  to  each  of  these  equal  quantities,  the  result  is 

x—24+6 
=30,  the  value  of  x  required.    ^ 

It  follows  from  (I.)  and  (II.)  that 


132  ALGEBRA. 

We  may  transpose  any  term  of  an  equation  from  one  member  to  the  other  ft) 
changing  the  sign  of  that  term.* 

We  may  change  the  signs  of  every  term  in  each  member  of  the  equation  with 
out  altering  the  value  of  the  expression.] 

If  the  same  quantity  appear  in  each  member  of  the  equation  affected  with  th 
tame  sign,  it  may  be  suppressed. 

Ill    Let  the  equation  be 

ax=b. 

Dividing  each  of  these  equals  by  a,  the  result  is 

b 
r=-,  the  value  of  x  required. 

So,  also,  in  the  equation 

6z=24. 

Dividing  each  of  these  equals  by  6,  the  result  is 

x=4,  the  value  of  a:  required. 

From  this  it  follows  that, 

When  one  member  of  an  equation  contains  the  unknoum  quantity  atone, 
affected  with  a  coefficient,  and  the  other  member  contains  known  quantities  only, 
the  value  of  the  unknown  quantity  is  found  by  dividing  each  member  of  the 
equation  by  the  coefficient  of  the  unknown  quantity 

TV.  Let  the  equation  be 

X-  =  b. 
a 

Multiplying  each  of  these  equals  by  a,  the  result  is 

x=ab,  the  value  of  x  required. 

So,  also,  in  the  equation 

x 

Multiplying  each  of  these  equals  by  6,  the  result  is 

x=144. 

From  this  it  follows  that, 

When  one  member  of  the  equation  contains  the  unknown  quantity  alone,  du 
tided  by  a  known  quantity,  and  the  other  member  contains  known  quantities 
only,  the  value  of  the  unknown  quantity  is  found  by  multiplying  each  member 
of  the  equation  by  the  quantity  which  is  the  divisor  of  the  unknown  quantity. 

V.  Let  the  equation  bo 

ax  dx     m 

b  c        n ' 

In  order  to  solve  this  equation,  we  must  clear  it  of  fractions ;  to  effect  this, 
reduce  the  fractions  to  equivalent  ones,  having  a  common  denominator  (Art. 
41),  the  equation  becomes 

aenr     been     bdnx     ban 
hi  it        l><  n        be n       ben  ' 
Multiply  these  equal  quantities  by  t ho  same  quantity  ben,  or,  which  u 

•  If  we  transpose  a  plus  term,  it  subtracts  this  term  from  both  members  ;  and  if  we 
transpose  a  minus  term,  it  tddi  this  term  to  both. 

t  This  is,  in  fact,  tho  sum.'  thing  as  transposing  ovory  term  in  each  member  of  the  eque 
tion,  or  multiplying  tliruuyhout  by  — 1. 


SIMPLE  EQUATIONS.  133 

aently  the  same  thing,  suppress  the  denominator  ben  in  each  of  the  fractions, 
the  result  is 

aenx — bcen=bdnx — bcm,  an  equation  clear  of  fractions. 

So,  also,  in  the  equation 

2x     3  x 

T-4  =  U  +  5- 
Reducing  the  fractions  to  a  common  denominator 

iOx     45      660      12x 
60~  —  60=="60~"^"60'' 

Multiplying  both  members  of  the  equation  by  60,  the  result  is 
40a: — 45=660-j-12.r,  an  equation  clear  of  fractions 
If  the  denominators  have  common  factors,  we  can  simplify  the  above  opera- 
tion  by  reducing  them  to  their  least  common  denominator,  which  is  done  (see 
Art.  44)  by  finding  the  least  common  multiple  of  the  denominators.     Thus,  in 
the  equation 

5x     4x         _7     13  r 

The  least  common  multiple  of  the  numbers  12,  3,  8,  6  is  24,  which  is,  there 
fore,  the  least  common  denominator  of  the  above  fractions,  and  the  equation 
will  become 

IOx     32.r     312     21      52x 
*24       "24  ~~ '~2A~2~i~~2A' 
Multiplying  both  members  of  the  equation  by  24,  the  result  is 

10.r — 32x — 312=21 — 52x,  an  equation  clear  of  fractions. 
Hence  it  appears  that, 

In  order  to  clear  an  equation  of  fractions,  reduce  the  fractions  to  a  common 
denominator,  and  then  multiply  each  term  by  this  common  denominator.     In  the 
fractional  terms  the  common  denominator  will  be  simply  suppressed. 
143.  From  what  lias  been  said  above,  we  deduce  the  following  general 

RULE  FOR  THE  SOLUTION  OF  A  SIMPLE  EQUATION  CONTAINING  ONE  UNKNOWB 

QUANTITY. 

1°.  Clear  the  equation  of  fractions,  and  perform  in  both  members  all  the  alge- 
braic operations  indicated. 

2°.  Transpose  all  the  terms  containing  the  unknown  quantity  to  one  member 
of  the  equation,  and  all  the  terms  containing  known  quantities  only  to  Hie  other 
\  member,  and  reduce  each  member  to  its  most  simple  form. 

3°.  We  thus  obtain  an  equation,  one  member  of  which  contains  the  unknown 
quantity  alone,  affected  with  a  coefficient,  and  the  other  member  contains  known 
quantities  only  ;  the  value  of  the  unknown  quantity  will  be  found  by  dividing  the 
member  composed  of  the  known  quantities  by  the  coefficient  of  the  unknown 
quantity. 

The  terms  containing  the  unknown  quantity  are  usually  collected  in  theirs* 
member  of  the  equation,  though  they  may  often  be  more  conveniently  col- 
lected in  the  second ;  the  second  being  afterward  written  as  the  first  member, 
and  the  first  as  the  second. 

Sometimes  an  equation  presents  itself  as  one  of  a  degree  higher  than  the 
first,  but  both  members  are  divisible  by  such  a  power  of  the  unknown  quan- 
tity as  to  reduce  the  equation  to" one  of  the  first  degree. 


134  •  :.;. 

In  other  cases,  clearing  an  equation  of  fractions  reduces  it,  by  the  cane, 
of  those  terms  which  contain  the  higher  powers  of  the  unknown  quantity,  to 
the  first  degree. 

A  proportion  containing  an  unknown  quantity  in  any  of  its  terms  can  be 
thrown  into  the  form  of  an  equation  by  multiplying  the  extremes,  and  also  th*» 
means,  and  setting  the  two  products  thus  formed  equal  to  each  other. 

j  \  AMPLE  I. 
Given,  19x+13   =59— 4x.  v 

Transposing.  19x+   4x=59 — 13. 

Reducing,  23x=46. 

Dividing  by  23,  x=2. 

Verification. — Substitute  2  for  x  in  the  given  equation,  it  becomes 
19x2  +  13=59—4x2,  or 

38+13=59— 8,  an  identity. 
Let  this  process  be  repeated  in  some  of  the  following  examples. 

EXAMPLE  II. 
_.  XX  XX 

Glven,  -_i  +  10  =     3-2+11. 

Reducing  to  least  common  denominator  12, 

2x     3x  Ax     6x 

12-12  +  10   =   12-12+1L 
Multiplying  both  members  by  12, 

2x— 3X+120   =  Ax— 6x+132. 
Transposing,    2x  — 3x— 4x  +  Gx=132  — 120. 
Reducing  x         =  12. 

EXAMPLE    III. 

5x+3  Ax— 10 

G>ven,  "^+7  =  -To~+1<)- 

Reducing  to  least  common  denominator  20, 
25x+15  8.r— 20 

~ir-+7  =  ^o-+10- 

Multiplying  both  members  by  20, 

25x+-15-|-140=  Sx— 20  +  200. 
Transposing,  25x—  8x=200— 20  — 15  — 140. 

Reducing,  17x=  25. 

25 

Dividing  by  17,  x=  — . 

1  \  \MII.I     IV. 

2x— 5      Tx+10  12x  — 10 

Given,  — ; — — — - —    =16 - . 

4  3  5 

Reducing  to  common  denominator, 

30x— 75      140x+200  144x— 120 

U0      —        GO        =1G—       qq        • 
Multiplying  both  members  by  60^ 

0  — 75— 140x— 200  =960— 144x+] 
Transposing,  30x— 140x+144x=960+  ?.'>  '+200+ 12a 


SIMPLE  EQUATIONS.  135 

iveducing,  34£=1355. 

_  1355 

Dividing  by  34,  x=         . 

Tt  is  unnecessary  to  write  the  common  denominator. 

EXAMPLE  V. 

12— 4.r     2aM-5  7.r+G0 

Given,  -I5 f-     =  3+-^~50. 

deducing  to  least  common  denominator,  10,  and  neglecting  it,  we  have' 

12— 4.r—4.r— 10   =30+   35.T+300— 500. 
Transposing,       — 4.t— 4.r— 35.r=30-}-300  —  12+10—500. 
Reducing,  —  43.r=  — 172. 

Changing  the  signs  of  both  members,* 

43.r=      172. 
Dividing  by  43,  x=         4. 

EXAMPLE  VI. 

Given,  ax-\-b  =.cx-\-d. 

Transposing,  ax — cx=  d — b. 

Simplifying,  (a — c)x=  d—b. 

d  —  b 
Dividing  by  (a — C),  x*= . 

EXAMPLE  VII. 

ax      CX  gX 

Reducing  to  a  common  denominator, 

adhx     bchx         .  bdgx 

i^+m+e=fx+-bdk+m' 

Multiplying  by  bdh, 

adhx-\-bchx-\-  bdeh=bdfhx-{-  bdgx-\-  bdhm. 
Transposing,   adhx-\-bchx — bdfhx—bdgx=  bdhm — bdeh. 
Simplifying,        (adh-\-bch — bdfh—bdg)x=bdhm — bdeh. 

bdhm — bdeh 
Dividing  by  coefficient  of  *,  X=adk+bch-bdfh-bdg 

bdh(m — e) 
adh-\-  bch — bdfh — bdg' 

EXAMPLE  VIII. 

x  dx 

Given,  —  1— +3aZ>=0. 

a  c 

Reducing  to  common  denominator  and  neglecting  it, 

ex — ac — adx-\-3a-bc=0. 

Transposing  and  simplifying,  (c — ad)x=ac — 3a:bc. 

ac(l — Sab) 
Dividing  by  coefficient  of  a:,  x=- ——5 — . 

Verification. 
ac(l—3ab) 

c — ad  acdll — 3ab) 

1— — 7^ 7rJ-+3ab  =  Q; 

a  c(c — ad)       ' 

*  Or  dividing  both  members  by  — 43,  gives  x=4. 


13G 
or 

or 


ALGEBRA. 


c(l—  3ab)  ad(l  —  3ab) 

■  1—  L4-3ab=0', 


c — at 


■ad 


c—3abc—c-\-ad  —  ad+3a"bd-\-3abc  —  3a"-bd=:Q. 


Given, 
Transposing, 


Given, 

Clearing  of  fractions 


EXAMPLE  IX. 

x+18=3x— 5. 

18+5  =3x— x 

23=2x 

23 
x ll1. 

EXAMPLE   X. 

a     b     d 

x     c'  e 
ace=bex-\-cdx 
ace=(be-\-cd)x 
ace 


X  = 


be-\-cd' 


EXAMPLE    XI. 


Given, 
Dividing  by  x, 


3x2— 10x=8x-|-x2. 
3x— 10  =8  -fx 
x=9. 


Given, 

Dividing  by  xm_1, 


Given, 


EXAMPLE   XII. 

xm=axm~l. 
x=a. 

EXAMPLE  XIII. 


axm— a' 


a 


■=a- 


rm— i 


xm  X" 

Multiplying  by  xm,         axm — a':=axm — a"x. 
Cancebng  axm  in  both  members, 


■a'  =  —a  'x .-.  x=— . 
a" 


Given, 


EXAMPLE  xiv. 
ad 


a:bx::c:d  •••  bcxz=ad  .-.  x= 


be 


144.  In  addition  to  tho  axioms  in  (Art.  141)  we  may  snbjox  the  fallowing: 

If  two  equal  quantities  be  raised  to  the  same  power,  the  results  will  be  equal. 

If  (lie  same  root  of  two  equal  quantities  be  extracted,  the  results  will  be  equal. 

Hence  any  equation  may  be  cleared  of  a  single  radical  quantity  by  trans- 
posing all  llie  other  tonus  to  the  opposite*  aide,  and  then  raisin:;  each  member 
to  the  power  denoted  l>\   the  index  of  the  radical.      If  there  be  more  than  on 
radical,  tho  operation  must  bo  repeated.     Thus  : 


SIMPLE  EQUATIONS.  137 

EXAMPLE   XV. 


Given,  -\/3.f-|-  7  =  10. 

Squaring  each  member  of  the  equation, 

3x+7  =  100. 
Transposing,  3x=l()0 — 7. 

Reducing,  and  dividing  by  3,  x=31. 

EXAMPLE  XVI. 


Given,  ■/4a:+2=  y/4x-\-5. 

Squaring  both  sides  of  the  equation, 

4^+2=4x4-10  \/4x"+25. 
Reducing,  —  10</lx=23. 

Squaring  both  sides,  400x=529. 

_529 

EXAMPLE   XVII. 

•/ar+28      t/x+38 


Given, 

Vi-f4        V-r+(j 

Clearing  the  equation  of  fractions, 

x+28  V^+6 -v/^+168=a:+38  /ar+4  V^+152. 
Transposing  and  reducing,  16=8  -\/x. 

Dividing  both  members  by  8,  2=    -J x. 

Squaring  both  members,  4=        x. 

EXAMPLE  XVIII. 


Given,  tya+x  =2^x34-5ax+62. 

Raising  both  members  to  the  mth  power, 

a+x  =  V^+5a.T+62. 

Squaring  both  members,   a2-\-2ax-{-x~=x'2-\-5ax-\-b2. 

Transposing  and  reducing,  — Zax  =b- — a2. 

Changing  the  signs,  3ax  =a2 — b2. 

a2—b2 
Dividing  by  3a,  x  = — - — . 


Given, 


EXAMPLE  XIX. 

yx—a*     27x— a 


-tyx—a  b 

Since  tyx  is  the  square  of  "tyx,  and  a-  is  the  square  of  a,  wo  can  perform 

the  division  indicated  in  the  first  fraction,  ;md  have  for  a  quotient 

2v/x—a 
2yx+a=  — £— , 

.:(b-iyy.v=-(b  +  l)a, 
(b  +  l)a 

.      2m/ ? V        '         > 

"  vx—  (6-1)' 
((b+l)ay° 

•'••r-V  b-i  ) 

(20)  Given  4x+36=5x+34.      "  Ans.  x=2. 

(21)  Given  4x— 12+3x+l=2x+4.  Ana.  x=3 


138  ALGEBRA. 

(22)  Given  .'.(/  +  ..  —  56+2=76— a+c+6,       Ans.  z=l2& — ia  +  r+4. 

(23)  Given  13J — — 2x— 8|.  Ans  x=9. 

(24)  Given  12} +  3x  —  r;_  —  ='- 5J.  Ans.x=l: 

a:     a-     i 

(25)  Given -+—=-+7.  Ansx=l2. 

(26)  Given  -+r.  +  -  =  13.  Ans.  x  =  12. 

x     x 

(27)  Given  x+- — -  =  4x — 17.  Ans.  x=ti. 

x+4 

(28)  Given  5 r^-=i— 3.  Ans.  x=7. 

3x— 5  2x— 4 

(29)  Given  *+  — —  =  12—— - — .  .  Ans.  x=5. 

x+1      x+3     x+4 

(30)  Given -^— +— 7— =— 1-+16.  Ans.  x  =  41. 
'                   3             4             5 

5x  4x 

(31)  Given  5x— — +12=— +  Ans.  1=12. 

_  x     4x  41x 

(32)  Given  7x+13?  — -=— — 8?+  — .  =9. 

2       5  o 

(33)  Given  6x— 7]  —  J-X+.10  — 5x— 24'=0.  Ans.  x=0,  or  8$. 

(34)  Given  4(5x+7—  -?-)  =  =  (3.r+9  — 4).  Ans.  x=— 1§. 

x+Jx+'x     20x— 25 

(35)  Given  -^-=— 0—  Ans.  x=2,V 

■7*  £  OP]  rp 

(36)  Given '——  +  Gx=—-r—.  Ans.  x=9. 

11 x     19 x 

(37)  Given  x+ — - — = — - — .  Ans.  x=5. 

2x+G  lis— 37 

(38)  Given  3x+— ~ =5  + —  .  Ans.  x=7. 

n,_4  18— i 

(39)  Given— r 2= - +  .r.  Ans.  X=4. 

:;,-_ll     ;,,•  —  :>     97—' 

(40)  Given  21+  =— — +—  ,— .  Ans.  r=9 

x— 1  5x+ 14       1 

(41)  Given  3x ; 1= — ;.  Ans.  r=7 

x      '  4  12 

x l      <">3 /•  4  +  /' 

(42)  Given -y-  +  :i-^—  =7-— J-  \ns.  x=8 

7.,+  r,     16+4i  ::<  + 

(13)   (liven — - — — - — +G= r— .  \ns.  x  =  ] 

'  :;  5        '  2 

3X+4      7,.  —  ::      ,,_n; 

(II)    (liven ; —  =  — ; .  Ans.  ,r=2 

■  >  2  4 

17—31       l;+2  7/+M 

(45)  Given  — — ^—  =5  —  Gx+  ina.  .;  =  . 

—3  20-  1     Gx— 8      kr — 1 

(46)  Given  X r-    +4  =  — —  —  <.  r=6 

.)  2  /  5 


SIMPLE  EQUATIONS.  13S 

4i  —  21              57  — 3x                5x— 9G 
v4?)  Given  ^-^  +  31+  — — =241-— liar.       Ans.  x=21 

fa+18     ,R      11— 3a:  13— a     21 -2x 

(48)  Given  -g 4*-- ^-=5x-48— ^— y^-. 

Ans.  x=10. 

(49)  Given  5.r+:iL^— =— 77—  +  •">•  Ans.  x=4. 

&.T     a7     a     ex  .  ad 

(50)  Given  __-— -^  Ans.  x=-^. 

5x— 1      3x— 2      llx— 3      13x— 15     8x— 2 

(51)  Given  23  +  ^-+  — —^~= 3 ~ 7— 

Ans.  x=9. 

1       3x— 13      12+7x  9+5x     llx  — 17 

(52)  Given  4x+-_~ __=7*_33— 55-  —  -g— 

Ans.  x=15. 

ace     (a  +  Z>)2x     ,  ,  .  a3e(c— d) 

(53)  Given  -7—  - — ! — —  bx=ae  —  3bx.  Ans.  x=,  2  ■  ^w- 

a+3x     7a — 5x  9x      x      5x 

(54)  Given______+3-T=^  +  65. 

39a&  — 14a3 
Ans.  x== 


■27ab  —  96  +  12" 
6x        (36c+ad)x       5a&    _(36e-^aa>     5a(2ft— a) 

(55)  Given  o^~  2a6(«  +  Z>)  ~~3c37z-  2a6(a-6)  "~    a3-&2    ' 

5a(2&  —  a) 

Ans.  x= — -j — • 

3c — a 

d — c 

(56)  Given  ax+c=bx+d.  Ans.  x=^— ^. 

4a' — 3ab 

(57)  Given  2ax— &x+2a&=4a3— ab— 3ax.  Ans.  x=    5a_i   • 

7a&  — 3a2 

(58)  Given  (3a— x)(a— 6)  +  2ax=4fc(a+x).  Ans.  g=  ■  ft    . 

11,  6c 

(59)  Given  -ax+-6x=c.  Ans.s=3     .  2&. 

x  dx  .  ac(l— 3afc) 

(60)  Given  — 1 +  3aZ>  =  0.  Ans.  x= — -_^     . 

a2X  aic — ac2+6cd —  '•'</ 

(61)  Given  ^—  +  c?c=6x—ac.  Ans.  x= j2_jc_a2 

ax  mx       ,  hcn-\-bdn 

(62)  Given  T-c=-+a\  Ans.  *=-—£_, 

«r  ex  8a63+4&3— 12a26 

(63)  Given  __+46=^^.  Ans.  *=3aB  +  ||6_flC+6c 

Six     x—b     bx—a*      x  4a9(as+a6— 6s) 

(64)  Given  — _— =-i-^_-.       Ans.  *=3a3_6a*6+ai3+66s 

(a+5)(x— &)  4a&— 5s  a''  —  6x 

(65)  Given     "^      ^3^-^p— 2x+~T- 

a^+S^fe +4a3&3— 6a63+26< 

Ans.  x_  i(4a-  +  2a6-26-) 


140  ALGEBRA. 

ax      b      ex     px      q      rx  kb-\-kq 

(66)  Given  — 4-  —  4-^-=— —  — — -r.       Ans.  x=  —  ■; ; — : : 

1     ;  m^mT  k      m     m      k  ka  —  kp  +  mc  +  rm 

W+q) 

k(a—p)-\-m{c-{-r)' 

x     ax     bx      ex     mx 
167)  Given  -  +  -  +  -  + -~+p. 

12Pk 
Ans.  x=: 


■V2(l—m)  +  2{3a-\--2b)  +  3t 

'  cq  —  rb 

(68)  Given  r{ax+b—c)=c{px+q  —  r).  Ans.  x=ra__     . 

x4-px—qx     mx—n  n(f]—p) 

(69)  Given  —^- -  = .  Ans.  x=     \       . 

v     '  p — q  m  m 

(70)  Given  (£m+i>)($a?-3r)Mfm+2p)(fx-7r). 

r(952m  +  4928p) 

9m+208/»      ' 


Ans.  x=- 


m*x     h?  frnrx—h-n—Bntgx 

(71)  Given — — 4-5rcx= 

x     '  n       g   '  bng 

4n/t2 
Ans.  x= 


5msg— 4m*+33fty 

13(5ax— 2236)      24(3ax— 20£Z>) 

(72)  Given ^    ,      = 7c*k     '     • 

(2041c— 4406?A-)6 
(455c — 648&)a 

13m — 7x     4m — x       m4-p 

(73)  Given ; + = kx. 

v     '  m-\-p         m — p        in — p 

limp  — 16m- +p* 
6ji — 8m-\-k(m: — p9)' 

3a6c      (2a  +  b)b-x        a-b2  bx 

(74)  Given  — ri+     /     ,  L  +/     ■  M3=to+7- 

3aibc(a-\-by--\-asbi 
Ans.  r=- 


■(3ac+i)(a+6)3— (2a  +  6)(a+i)6« 


a2— Six  6&x— 5a2     £x+4a 

(75)  Given  ax— — <z&2=6x-| . 

1  a  '        2a  4 


■\ab'— 10a 

ADS-  a'=    4«-3fc   * 
</-6 


(76)  Given  rtxc4-tx=c.f2-f-rfx.  Ans.     = 


</  —  c 


D4-B 

(77)  Given  Axm4-Bxm-1  =  Cxm— Dxm-'.  Ans.  x=     _     . 

14a3— 2a26x  21a34-5a26x 

(78)  Given — 31^= -^ \-20c\ 

v     '  17mx  33mx        ' 

105a3 

Ans.  X=: 


'151a-7>  +  28611c*m 


4m(K2— 5x2)  r»n{«*  —  2r)  2K9 

(79)  Given — =7mp+-    — .  Ans.  xsr^g— p. 

24x*  5X8 

(80)  Given  3-^=^^.  Ans.x=3]>. 


181 
<82 

(83 
(84 
(85 

(86 

(87 
(88 
(89 
(90 
(91 
(92 

(93 
(94 

(95 

(96 

(97 

(98 

(99 


SIMPLE  EQUATIONS.  141 

_  axa  mx"  bm — ap 

Given  j— — = — ; .  Ans.  x= . 

b-\-cx     2)~\~(lx  a<l — cm 

x 
Given  12 — x:-::4:l.  Ans.  x=4. 

5x+4   18— x 
Given — ^ — : — - — ::7:4.  Ans.  x=2. 

2    '        4 

Given  2© :1::  1:3.1416.  Ans.  0=0.1591. 

b  7ad> 

Given  a: t::-: 7c.  Ans.  t 


c 


b   * 


c 

Given  r :  1 : :  c :  3.1416.  Ans.  r =—  rr=. 

0.141b 


Given  -v/4x+16  =  12.  Ans.  x=32. 

Given  V2x+3+4=7.  Ans.  x=12. 

Given  -/l2+x=24-  y/x.  Ans.  x=4. 

Given  -^+40=10  —  y/x.  Ans.  x=9. 

Given  y/x — 16=8 — y/x.  Ans.  x=25. 

Given  V^— 24  =  y/x— 2.  Ans.  x=49. 

_          _  25a 

Given  y/x — a=  yfx— \yf  a.  Ans.  x=tt^-. 

-        9 

Given  y/bx  ■v/^+~=  y/bx-\-2.  Ans.  x=— . 


■1 


_  {b—a\ 

Given  y/4a-\-x=2y/b-\-x — y/x.  Ans.  x=       ,  . 

(6— of 

Given  x-}-a  +  y/2ax-\-x~b.  Ans.  x== — —? — 

x — ax      y/x 


j. 


Given  — =-= .  Ans.  x=; 

y/x         x  1~a 

V^+28      V^+38 

Given  — z= = — = •  Ans.  x=4 


"v/x-f-4        y/x-\-Q 

.,.         y/x4-2a      y/x+4a  1  ab  \* 

Given  — — r= — — .  Ans.  x=( j)  . 

y/x+b       y/x+Zb  \a  —  bj 

3x— 1  V3x— 1 

(100)  Given  —== =  1+ o •  Ans.  x=3 

y3x-{-l 

ax—b*             y/ax—b  1/.  ,     c3  \« 

(101)  Given  =H--^7 ■  An8>  ;r=a(6+^i ) 

1 t2— 4a* 

(102)  Given  x=     a^+xy/b^+x^—a.  Ans.  x= — ^— 

25 

(103)  Given  y/b+x4-  y/x=—z==.,  Ans.  x=4 

V5+x 

<104)  Given  a/x+  y/x—Jx—  y/x=-yj =•  Ans.  ^=r^. 


1   1     /r~   n~  9 

(105)  Given -+-=V^+V^+^- 


Ans.  x=2a. 


1  I I  ALGEBRA. 


46 

(106)  Given  i/lQx+3=7.  Ans.  x=—. 

(107)  Given  y/.c  —  3:7=10  —  V*  Ans.  x=81. 

— 9  V^-i' — 3 

(108)  Given  —= —  1  =  ^—- .  \  ns.  x=5. 

V5x+3 

(109)  Given  h  ^/a.v—b=k  Vcx+dx—f.  Ans.  x= — — — J— 

an? —  (c-\-d)l<? 

\fa-\-x-\-  y/ a — .r  2ay/m 

(110)  (liven  — = =  -Jm.  Ans.  x=— . 

V«+.r—  V«— x  1+m 

(111)  Given  V«2+c=y  ,?**»,•  Ans.  x=     s-  - 9. 

'  •     '  Vrf(.r-|-9)  ayu--|-c 

„  mi    , nwj  (nr-X-mq)(nr — mq\ 

112)   (iiven  —  V/;-x-+fy2+-L- =/r.         Ans.  x=       ^  J'K ^. 

'  n      J        '  *    '      w  2>nnpr 

When  an  equation  can  never  he  verified,  whatever  value  we  put  in  the 
■  of  the  unknown  quantity,  it  is  said  to  be  impossible  ;  and  when  an  equa- 
tion is  always  verified,  whatever  value  be  put  for  the  unknown  quantity,  it  is 
6aid  to  be  indeterminate. 

CASES  OF  IMPOSSIBILITY  AND  [NDEffERMINATION  IX  EQUATIONS 

OF  THE  FIRST  DEGREE. 

I.  Problem. — To  find  a  number  such  that  the  third  of  it,  augmented  by  75, 
and  five  twelfths  of  it,  diminished  by  35,  shall  make  three  quarters  of  it,  added 
to  49. 

The  equation  is 

x  5x  3x 

-+75  +  --35=1+49,  [1] 

x     5x     3x 

•••  3+r2-T=9 

.-.  4x-\-5x— 9x=108 

.-.  0  =  108. 

An  absurdity.     There  is,  therefore,  no  value  of  x  which  can  satisfy  the 

equation  [1]. 

The  impossibility  may  be  rendered  evident  in  the  equation  [1]  itself  by  re 

ducing  the  similar  terms  in  the  firsl  member;   thus, 

3x  3x 

T+40=T+49. 

It  is  evident  that  the  two  members  will  always  differ  by  9,  whatever  bo  the 
value  of  x. 

II.  Problem. — To  find  a  number  such  that,  adding  together  the  half  of  it  in 
creased  by  10,  two  thirds  of  it  increased  by  "J'',  and  live  sixths  of  it  diminish 
ed  by  34,  the  sum  shall  lie  equal  to  twice  the  Bxceas  of  this  number  over  5. 

r+io    2(x-f20)    r>(./— ::t) 

...  3x+30  +  4x+80  +.r»x  —  170=12x— 60 
...  3,-^.  ir-f  ;,.,—  1  ■.'./•=:  170  —  30  —80  —60 
I*.  <.,  0  =  0. 
The  unknown  x  is,  therefore,  altogether  indeterminate;  that  is  to  say,  it 
way  be  taken  equal  to  2  or  ■''.  or  any  number  whatever. 


SIMPLE  EQUATIONS.  143 


ON  THE  SOLUTION  OF  SIMPLE  EQUATIONS,  CONTAINING  TWO  OR 
MORE  UNKNOWN  QUANTITIES. 

145.  A  single  equation,  containing  two  unknown  quantities,  admits  of  an  hi' 
finite  number  of  solutions;  for  if  we  assign  any  arbitrary  value  to  one  of  Che 
unknown  quantities,  the  equation  will  determine  the  corresponding  value  of 
the  other  unknown  quantity.  Thus,  in  the  equation  y=x-\-lQ,  each  value 
which  we  may  assign  to  a;  will,  when  augmented  by  10,  furnish  a  correspond- 
ing value  of  y.  Thus,  if  x=2,  J'  =  l~  ;  if  x=3,  y=13,  and  so  on.  An  equation 
of  this  nature  is  called  an  indeterminate  equation,  and  since  the  value  of  y  de- 
pends upon  that  of  x,  y  is  said  to  be  n  function  of  x. 

In  general,  every  quantity,  whose  value  depends  upon  one  or  more  quantili/s, 
is  said  to  be  a  function  of  these  quantiti   :. 

Thus,  in  the  equation  y=ax-\-b,  we  say  that  y  is  a  function  of  x,  and  that 
y  is  expressed  in  terms  of  x,  and  the  known  qualities  a,  b. 

If,  however,  we  have  two  equations  between  two  unknown  quantities,  and 
if  these  equations  hold  good  together,  then  it  will  be  seen  presently  that  wo 
can  combine  them  in  such  a  manner  as  to  obtain  determinate  values  for  each 
of  the  unknown  quantities ;  that  is  to  say,  each  of  the  unknown  quantities  will 
have  but  a  single  value,  which  will  satisfy  the  equations.  The  equations  in 
this  case  are  called  determinate. 

In  general,  in  order  that  questions  may  admit  of  determinate  solutions,  we 
must  have  as  many  separate  equations  as  there  are  unknown  quantities ;  a 
group  of  equations  of  this  nature  is  called  a  system  of  simultaneous  equations. 

If  the  number  of  equations  exceed  the  number  of  unknown  quantities,  un- 
less the  equations  in  excess  conform  to  the  values  of  the  unknown  quantities 
determined  by  the  others,  the  equations  are  said  to  be  incompatible.  Thus, 
if  we  have  x-\-y  =  l0  and  x — y  =  G,  the  only  values  of  x  and  y  which  will  satisfy 
both  these  equations  are  8  for  x,  and  2  for  y.  Now,  if  we  were  to  add  an 
other  equation  to  these,  it  must  conform  to  these  values,  and  could  not  be 
written  in  any  form  at  pleasure.  Thus,  wo  might  for  a  third  equation  say 
zy =16  ;  but  we  could  not  write  .r?/=100,  for  this  third  equation  would  be  in- 
compatible with  the  other  two.* 

*  Equations  may  be  incompatible  when  the  number  does  not  exceed  the  number  of  un- 
knowns, as  the  following  problem  will  show : 

A  sportsman  was  asked  how  many  birds  he  had  taken.  He  replied,  if  5  be  added  to  the 
third  of  those  I  took  last  year,  it  will  make  the  half  of  the  number  taken  this  year.  But  if 
from  three  times  this  last  half  5  be  taken,  you  will  have  precisely  the  number  taken  last 
year.     How  many  did  he  take  in  each  year  ? 

Let  x=  the  number  this  year,  and  y=  the  number  last  year. 

x     ii  .             3x 
-=-+5,  y— 5. 

2      3  ~  '  J       2 

Substituting  in  the  first  the  value  of  y  in  the  second, 

x       x     5     , 
2       2     3  ~ 
.-.  3.1- — 3j=30 — 10 
0  =20; 
an  absurd  equality,  whence  we  conclude  that  there  exist  no  values  of  .r  and  y  which  satisii 
the  two  equations. 

This  is  because  the  conditions  of  the  problem  are  inconsistent  with  each-  other.  When, 
however,  the  two  equations  are  derived  from  the  same  problem,  and  its  conditions  <u"o  not 
rontradictory,  values  for  x  and  y  wiV  always  be  found  to  satisfy  them. 


144  ALGEBRA. 

146.  In  order  to  solve  a  system  of  two  simple  equations  c  '■■> '•  ing  two  un- 
known quantities,  we  must  endeavor  to  deduce  from  them  a  single  equation 
containing  only  one  unknown  quantity;  we  must,  therefore,  make  one  of  the 
unknown  quantities  disappear,  or,  as  it  is  termed,  we  must  eliminate  it.  The 
equation  thus  obtained,  containing  one  unknown  quantity  only,  will  give  the 
value  of  the  unknown  quantity  which  it  involves,  and,  substituting  the  value  of 
this  unknown  quantity  in  either  of  the  equations  containing  the  two  unknown 
quantities,  we  shall  arrive  at  the  value  of  the  other  unknown  quantity. 

The  process  which  most  naturally  suggests  itself  for  the  elimination  of  one 
of  the  unknown  quantities,  is  to  derive  from  one  of  the  two  equation?  an  ex- 
pression for  that  unknown  quantity  in  terms  of  the  other  unknown  quantity, 
and  then  substitute  this  expression  in  the  other  equation.  We  shall  see  thai 
the  elimination  may  be  effected  by  different  methods,  which  are  more  or  less 
simple  according  to  the  nature  of  the  question  proposed. 

example  I. 
Let  it  bo  proposed  to  solve  the  system  of  equations 

y— *=  c (i)? 

y+x=\2 (2)  $ 

147.  First  Method. — From  equation  (1)  we  find  the  value  of  y  in  terms 
of  x,  which  gives  i/=x-f-6  ;  substituting  the  expression  x-\-G  for  y  in  equation 
(2),  it  becomes  x+G-f-x=12,  from  which  we  find  the  determinate  value  x=3 ; 
since  we  have  already  seen  that  y=x-\-G,  we  find  also  the  determinate  value 
y=s3+6  or  9. 

Thus  it  appears,  that  although  each  of  the  above  equations,  considered  sep- 
arately, admits  of  an  infinite  number  of  solutions,  yet  the  system  of  equations 
admits  only  one  common  solution,  x=3,  y=9. 

148.  Second  Method. — Derive  from  each  equation  an  expression  for  y  in 
terms  of  x,  we  shall  then  have 

2/=  a: -f-6 
y=12—  x. 

These  two  values  of  y  must  be  equal  to  one  another,  and,  by  comparing 
them,  we  shall  obtain  an  equation  involving  only  one  unknown  quantity,  viz., 

x+6=12  —  x. 
Whence 

x=3. 
Substituting  the  value  of  x  in  the  expression  y=x-\-G,  we  find  y=9. 
The  substitution  of  3,  the  value  of  x,  in  the  second  expression,  2/  =  12 — x, 
leads  necessarily  to  the  samo  value  of  y  ;  thus,  12  —  3=9,  for  we  derived  the 
value  of  x  from  the  equation  x-f-G  =  12 — X. 

149.  Third  Method. — Since  the  coefficients  of  y  aro  equal  in  the  two 
equations,  it  is  manifest  that  wo  may  eliminate  y  by  subtracting  Hie  tico  equa- 
tions from  each  other,  which  gives 

(y+.v)-(y—v)  =  ii-6. 
Whence 

2x=6 
*ss3. 
Having  thus  obtained  tho  value  of  r,  we  may  deduce  that  of  y  by  making 
r=3  in  either  of  tho  proposed  equations;  wo  can,  however,  determine  the 


SIMPLE  EQUATIONS.  145 

•alue  of  y  directly,  by  observing  that,  since  the  coefficients  of  x  in  the  proposed 
equations  are  equal,  and  have  opposite  signs,  we  may  eliminate  a:  by  adding 
the  two  equations  together,  which  gives 

Whence 

7/  =  9. 

If  we  examine  tho  three  above  methods,  wo  shall  perceive  that  they  con- 
sist in  expressing  that  the  unknown  quantities  have  the  same  values  in  both 
equations. 

These  methods  have  derived  their  names  from  the  processes  employed  to 
effect  the  elimination  of  the  unknown  quantities. 

The  first  is  called  the  method  of  elimination  by  substitution. 

The  second  is  called  the  method  of  elimination  by  comparison. 

The  third  is  called  the  method  of  elimination  by  addition  and  subtraction. 

Tho  rule  for  the  first  is  to  find  the  value  of  one  of  the  unknown  quantities  in 
one  of  the  equations,  and  substitute  it  in  the  other  equation. 

For  the  second,  is  to  find  the  value  of  the  same  unknown  quantity  in  each  of 
the  two  given  equations,  and  set  these  values  equal. 

And  for  tho  third,  is  to  make  the  coefficient  of  the  unknown  quantity  to  be 
eliminated  the  same  in  the  two  equations,  and  add  or  subtract  as  the  case  may 
require.  Add,  if  the  signs  of  the  equal  terms  are  different,  and  if  they  are 
alike,  subtract. 

By  either  of  these  rules  a  single  equation,  containing  but  one  unknown  quan 
tity,  is  obtained. 

EXAMPLE  II. 

Take  the  equations 

2x+3y=13 i (1)  > 

5.r-j-42/=22 (2)  S 

1°.  Eliminating  by  substitution. 
From  equation  (1),  we  find 

13— 2x 

Substituting  the  value  of  y  in  terms  of  x  in  equation  (2),  it  becomes 

13  —  2x 
5z+4x — 3 — =22; 

*n  equation  containing  x  alone,  which,  when  solved,  gives 

x=2. 
This  value  of  x,  substituted  in  either  of  the  equations  (1)  or  (2),  gives 

y=3. 

2°.  Eliminating  by  comparison 

13— 2x 
From  equation  (1)  2/r= — . 

22— 5x 
From  equation  (2)  y= — - — . 

13— 2x     22— 5.T 
Equating  these  values  of  y,  — - — = — -z — ;  an  equation  containing xonry 

K 


146  ALGEBRA. 

Whence 


-O 


Substituting  this  value  for  x  in  either  of  the  preceding  exj  ressions  lor  y 
we  find 

2/  =  3- 
3°.  Eliminating  by  subtraction. 

In  order  to  eliminate  y,  we  perceive  that  if  we  could  deduce  from  the  pro- 
posed equations  two  other  equations  in  x  and  y,  in  which  the  coefficients  of  y 
should  be  equal,  the  elimination  of  y  would  be  effected  by  subtracting  one  of 
these  new  equations  from  the  other. 

It  is  easily  seen  that  we  shall  obtain  two  equations  of  the  form  required  if 
we  multiply  all  the  terms  of  each  equation  by  the  coefficient  of  y  in  the  other. 
Multiplying,  therefore,  all  the  terms  of  equation  (1)  by  4,  and  all  the  terms  of 
equation  (2)  by  3,  they  becomo 

8ar+12y=52 
15*4-12^=66. 
Subtracting  the  former  of  these  equations  from  the  latter,  we  find 

7x=14. 
Whence 

z=2. 
In  like  manner,  in  order  to  eliminate  x,  multiply  the  first  of  the  proposed 
equations  by  5,  and  the  second  by  2,  they  will  then  become 

10x+15?/=G5 
10*4-  8y=44. 
Subtracting  the  latter  of  these  two  equations  from  the  former, 

7y=2l. 

Whence 

y=3. 

In  order  to  solve  a  system  of  three  simple  equations  between  three  unknown 
quantities,  we  must  first  eliminate  one  of  the  unknown  quantities  by  one  of  the 
methods  explained  above ;  this  will  lead  to  a  system  of  two  equations,  con- 
taining only  two  unknown  quantities ;  the  value  of  these  two  unknown  quan 
tides  may  be  found  by  any  of  tho  methods  described  in  the  last  article,  and 
substituting  tho  value  of  these  two  unknown  quantities  in  any  one  of  the  original 
equations,  wo  shall  arrive  at  an  equation  which  will  determine  the  value  of  the 
third  unknown  quantity. 

EXAMPLE  III. 

Take  the  system  of  equations 

3.r4-2y+  Z=16 (\)\ 

2z4-2y+2z=18 (2)  C 

2x+2y+  r=14 (3)  ) 

1°.  Eliminating  by  substitution. 
From  equation  (1),  we  find 

r  =  lG  —  3.r— 2y (1). 

Substituting  this  value  of;  in  equations  (2)  and  (3),  thoy  become 
2x+ 2y+2(16— 3x— 2y)=18  .  .  .  ('»)  ? 
2ar+2y4-  (16— to— 2y)=14  .  .  .  (6)  S 
these  last  two  equations  contain  x  and  y  only,  and.  if  treated  according  to  any 
of  tho  ubove  methods,  will  give  us 

x=2,  v=3. 


SIMPLE  EQUATIONS.  147 

Substituting  these  values  of  x  and  y  in  any  one  of  the  equations  (  ),  (2),  (3), 
4),  wo  find 

2°.  Eliminating  by  comparison. 

In  order  to  eliminate  z?  derive  from  each  of  the  three  proposed  equations  a 
value  of  z  in  terms  of  2  and  y  ;  we  then  have 

2  =  1G  — 3.c— 2y 
2=  9 —  x —  y 
2=14 — 2.r — 2y  ; 
equating  the  first  of  these  values  of  z  with  the  second  and  with  the  third  In 
succession,  we  arrive  at  a  system  of  two  equations  : 

16— 3x— 2y=  9  —  x—  yt 
16  —  3x—2y  =  li—2x—2y  \ 
containing  x  and  y  only  ;  these  equations  give 

x=2,  2/=3; 
these  values  of  x  and  y,  when  substituted  in  any  of  the  three  expressions  for 
r,  give 

2  =  4. 

3°.  Eliminating  by  subtraction. 

In  order  to  eliminate  z  between  equations  (1)  and  (2), 

3x+2y-\-  z_i6 
2x+2y+2z  =  18; 
we  perceive  that,  in  order  to  reduce  these  equations  to  two  others  in  which 
the  coefficients  of  z  shall  be  the  same,  it  will  be  sufficient  to  divide  the  two 
members  of  the  second  equation  by  2,  for  we  thus  have 

Subtracting  this  from  the  first  equation, 

3.r+2?/+2  =  16, 
we  find  an  equation  between  two  unknown  quantities, 

2x+y=7 (a). 

Tn  order  to  eliminate  z  between  equations  (1)  and  (3), 

3a;+2?/+2=16 
2x+2y+z  =  U. 
Subtract  the  latter  from  the  former,  which  gives 

the  substitution  of  this  value  of  x  in  equation  (a)  gives 

y=3, 
and  the  substitution  of  these  values  of  x  and  y  in  any  of  the  proposed  equa 
tions  gives 

2  =  4. 

The  particular  form  of  the  proposed  equations  enables  us  to  simplify  the 
above  calculation ;  for  if  we  subtract  equation  (3)  from  equations  (1)  and  (2) 
in  succession,  we  have 

(3x+2y+  2)— (2x+2y+z)=16— 14,  whence  xz=2 
(2x+2y+2z)  —  (2x+2i/+2)  =  18— 14,  whence  2=4  ; 
and  substituting  these  values  of  .r  and  z  in  any  of  the  proposed  equations,  we 
find 

y=3. 


U8  ALGEBRA. 

In  order  to  solve  a  system  of  four  equal  ions  between  four  unknown  qua»ti 
we  reduce  this  case  to  the  last  by  eliminating  one  of  the  unknown  quantities. 
We  thus  arrive  at  ;i  Bystem  of  three  equations  between  three  unknown  quan- 
tities, from  which  the  value  of  these  three  unknown  quantities  may  be  found. 
Substituting  these  values  in  any  one  of  the  equations  which  involve  the  othe; 
unknown  quantity,  we  deduce  iVom  it  the  value  of  that  unknown  quantity. 

EXAMPLE   IV. 
Take  the  system  of  equatii 

*■'■+.'/  +  =  +  «=" (I)) 

x+y+z-  t=  4 (-2) 

.(•-p-//  — Z  +  LV  =  11 (3) 

■'—'/+-+  (4)  J 

The  first  equation  give 

t=U—x—y—~     (5). 

Substituting  this  exp  n  i'w  t  in  the  three  other  equations,  we  find 

x+  V+  z=;  9 (G) 

x+  y+3z=n (7) 

*+%+  2=12      (8). 

In  order  to  solve  these  three  equations  betwt  ;,  we  find  from  the 

first 

r  =  9_.r-v/ (9)  ; 

and  substituting  this  value  of;  in  the  two    ther  equations,  they  become 

*+2/=-5 (10) 

2/=3 (11) 

Whence  x=2 (12). 

Substituting  the  values  of  x  and  y  in  equation  (6),  we  find 

z=4 (13). 

Substituting  these  values  of  a:,  y,  z  in  any  of  the  first  five  equations,  wo  find 

t—b. 
We  can  arrive  at  the  same  result  more  simply  by  subtracting  equation  (1) 
from  the  three  following  in  succession  ;  we  shall  thus  find 

2i=14  —  4,  2z— *=14— 11,  -J//— 2<r=14— 18; 
the  first  of  these  three  new  equations  gives  i=5  ;  this  value  of/,  substituted 
in  the  two  other  equations,  gives  z  =  4,  y  =  3  ;  and  substituting  these  values  of 
y,  z,  t  in  any  one  of  the  original  equations,  wo  End  ..=  •.'. 

By  following  a  process  of  reasoning  analogous  to  the  above,  wo  shall  be  able 
to  resolve  a  system  of  any  number  of  equations  of  the  first  degree,  provided 
there  be  as  many  equations  as  unknown  quant i' 

It  frequently  happens  that  each  of  the  proposed  equations  do  not  involve  ah 
the  unknown  quantities.  In  this  case,  a  little  dexterity  will  enable  us  to  effect 
the  elimination  very  quickly. 

KXAMPLK   V. 

Take  tho  system  of  equations 

'/—  ::-y+2:  =  13 (1) 

41  —  Jr=30 (-'I 

47/4-~':  =  14 (3) 

5i/ +3*  =32 (4) 

Upon  examining  theso  equations,  wo  perceive  thai  the  elimination  of  z  be 


SIMPLE  EQUATIONS.  149 

tweeu  equations  (I)  and  (3)  will  give  an  equation  in  x  and  y,  and  "that  the 
elimination  of  t  between  equations  (2)  and  (1)  will  give  a  second  equation  in 
r  and  y.     These  two  unknown  quantities  may  thus  be  easily  determined  : 
The  elimination  of  z  between  (1)  and  (3)  gives     ....     7y — 2x=   1 
The  elimination  of  t  between  (2)  and  (4)  gives     ....  20?/ -f  6.r=: 
Multiply  the  first  of  these  equations  by  3,  and  then  add 

them,  wo  have 41i/=41 

Whence y=   1 

Substituting  the  value  of  y  in  7y—2x=zl,  we  have  .     .     .  x=  3 

Substitute  this  value  of  x  in  (2),  we  have At — 6=30 

Whence '=9 

Finally,  the  substitution  of  the  value  of  y  in  (3)  gives     .     .  ~=  5 

The  following  general  rule  may  be  given  for  a  system  of  any  number  of 
equations  : 

Eliminate  one  of  the  unknown  quantities  by  combining  the  first  equation 
with  each  of  the  others,  or  by  combining  them  all  in  any  way  in  separate 
pairs.  The  number  of  equations  and  of  unknown  quantities  is  thus  made  one 
less.  Proceed  with  these  in  the  same  way  till  there  is  but  one  equation  and 
one  unknown  quantity.  Having  found  the  value  of  this,  substitute  it  in  a  pre- 
ceding equation  containing  but  two  unknown  quantities,  which  will  then  have 
but  one,  whose  value  may  be  found.  Substitute  the  values  of  the  two  un- 
known quantities  thus  found  in  an  equation  immediately  preceding,  containing 
only  three,  and  so  on,  till  all  the  values  of  the  unknown  quantities  are  obtained. 

We  have  seen  in  the  method  of  elimination  by  subtraction  that,  in  order  to 
render  the  coefficients  of  the  unknown  quantity  the  same  in  both  equations, 
we  must  multiply  each  of  the  equations  by  the  coefficient  of  the  unknown 
quantity,  which  it  is  required  to  eliminate,  in  the  other.  If  the  coefficients  of 
the  unknown  quantity  have  a  common  factor,  this  operation  may  be  simplified; 
thus 

EXAMPLE  VI. 

'T'ake  the  system  of  equations 

12.r+32?/=340 (1)  > 

8x+  24?/ =2.34 (2)  S 

In  order  to  render  the  coefficients  of  y  equal,  observe  that  32  and  24  have  a 
common  factor,  8  ;  it  will  suffice  then  to  multiply  equation  (1)  by  3,  and  equa- 
tion (2)  by  4  ;  they  then  become 

36.?;-4-9G.y=1020 
32.1- -f96?/=l  016. 
Subtracting  the  latter  from  the  former, 

4.r=4 
.r=l. 
Again,  in  order  to  elimiuate  x,  since  12  and  8  have  a  corirnon  factor,  4,  it 
will  suffice  to  multiply  equation  (1)  by  2,  and  equation  (2)  by  3  ;  we  then  have 

24ar+64y=680 
24x+72y=762. 
Subtracting  the  former  of  these  two  equations  from  the  latter,  we  have 

82/=82 
2/=10{. 


150  ALGEBRA 

(7)  Given  x+y  =  15 (1) 

x-y=  7 (2) 

Ans.  x=ll,  7/=  4. 

(8)  Given    x+y  =  10 (1) 

2x—3y  =  5 (2) 

Ans.  x=7.  y  =  3. 

(9)  Given  2x+3y= 13 (1) 

5x+4y=22 (2) 

Ans.  x =2,  1/ =  3 

(10)  Given    x=4y (1) 

2x-f  3i/=44 (2) 

Ans.  x=lG,  2/  =  4. 

(11)  Given  2x+3?/  =  70 (1) 

4x+5y=l30 (2) 

Ans.  a-=20,  y  =  li) 

(12)  Given  3x—5t/=13 (1) 

2.r+72/  =  81 (2) 

\ns.  .r=16,  i/^7. 

(13)  Given  ll.r-f  3?/  =  100 (1) 

4x—7y=     4 (2) 

Ans.  .r=8,  y=  1 

(14)  Given  |+|=7 (1) 


2  '  3 

x     y 

-4-- 
3^2 


x     y 


Ans.  :r=6.  J/= 12. 


(15)  Given  '^+72/=99 (1) 

f+7s=51 (2) 


Ans.  arr=7,  ?/= 1 4 


7u 

(16)  Given     3<+  — =22 (1) 

°t 
11m— ^=20 (2) 

.  /  =  •">.   l/  =  2. 

(17)  Given.r-f  l:?/::5:3 (1) 

7_L;r     5_7/      42     2.r  — 1 
4  2     -12  4        ^ 


Ans.  x=4,  ?/  =  3 
ja      9s 

g"+Io' 


2r     4s 
(18)  Given  -+— =G4 (1) 

-4-—  =77 (0) 

:    i    in  v    ' 


Ans.  r=60,  s  =  30 

(19)  (liven    5p+fcr=131J (1) 

13/)—  ff=142j (2) 

.p=16J$$,  r  =  : 


SIMPLE  EQUATIONS.  161 

(20)  Given  6?*— 14^=5^+119? (1) 

7^+140=2^ (2) 

Ans.  *=— 24.07,  V=— 14.24 

(21)  Given  9x=4.r' (1) 

.r+.r'=2G  (2) 

Ans.  x  =  8,  a:'=18. 

(22)  G^T^T  =  8i  +  Z (1) 

21z1  +  28z2=334       (2) 

Ans.  z1=61T,TS9,  z2  =  —  33i|f, 

V  +  3     n  ti\ 

(23)  Given  2a:— ^-=8 (1) 

*_*=!_*»-*£ » 

Ans.  ar=5,  y=5. 

?/-8     3.r+4y+3     2.r+7-y 

(24)  Given    5+— g-  = -Jq  J5         ■  •  •  •  W 

7.r+6     9y+5.r-8_    .r+y 
11     ~         12  4      l  ' 

Ans.  .r=7,  y=9. 

,25)  Given  (x+5)(7/+7)  =  (x+l)(y-9)  +  112.  ...  (1) 

2:r+10=3y+l (2) 

Ans.  x=3,  y=5. 

o-j.            •;/                  3v      1 
(26)  Given— —4+--+.r=S—j+Y2 U) 

2_-+2=|— 2z+6 (2) 

Ans.  x=2,  y=7 

x_2     10— .r     y— 10 
(27)  Given    — g —^—=—j— (1) 

2.V+4     2.r+y     g+13 

3     "~      8     "~      4        V~; 

Ans.  x=7,  y=10 

2y     8.r— 2  4+?y     a;— y  . 

z:3y::4:7 (2) 

Ans.  £=12,  y=7. 

4y 
15x4-  — 
3y— 2+.r     n  ,              3 
(29)  Given  x ^-j^ =1  +  — 33— (!) 

3.r+2y     y-5     llx+152     3y+l 

6      ""    4    "       12  2        *  *      '  K~> 

Ans.  x=8,  y=9 


25+51/     7x— 6      ,A      3x— 10+7y 
(30)  Given  1  +  — p^ 3— =10 jg-     -  •  •  U) 

.(2) 

Ans.  .r=3,  y=7 


__:D,.____::l:8 (~) 


152 


ALGEBRA. 


Ax     by     9 
(31)  Given*  —  +  4=-  —  1 
'  x2      y"      y 

5     4_7     3 

x'y     x'2 


(1) 
(2) 


\32)  (Ihi'i)    5x+7#=43    . 
llx+<ty=G9    . 

(33)  Given  8x— 21y=  33 
6x+35?/=177 


Ans.  x=4,  ?/=2 

."CO  J 

•  (2)  S 

Ans.  x=3,  y=4 

.(2)5 

Ans .  .r = 1 J ,  y  =3. 


2x  v  3?/      1 

(34)  Given  T-4+^+x=8-^+- 

y     x  1 

i-2+2  =  6-2*+6- 


(1) 


(35)  Given  x- 


3x+5.?/ 


fl7=5y-| 


4x+7 


•  (2) 

Ans.  x=2,  ?/=7 

•  (1)1 


(2) 


17       '  *  l       3 

22— Gy     5x— 7_x-fl     8?/+5 

3  IT~~    6     ~ ~T8~ 

Ans.  .r=8,  y=2. 

36)  Given  ax+  by=c '.  .  .  (1)  ) 

fx+gy=h    .  .  (2)  $ 

—  fr/i  a/t — c/" 

Ans.  x==- ' 


(37)  Given  x-\-y=s  . 
x — y—d 


'ag—bf  y     ag—bf 

(1) 
(2) 


s-\-d  s — d 

Ans.  x=— — ,  y=—z~ 


(38)  Given    x-\-y=s (1) 

bx=ay (2) 


as  bs 

Ans.  x= — — ,  2/=- 


■a+b,:f—a+b 

(39)  Given  ax+by=c (1) 

mx — ny=d (2) 

nc-\-lnI  vtc — ad 

Ans.  x= j — T,  w= - r. 

na-\-mb   J      na-\-mb 

(40)  Given  7ax=4& (1) 

2cx  +  3dy=4c (2) 

46  28ac— 8bc 

Ans.  x=— ,  y= — - — -. — . 
la  J  2lad 

(41)  Given  bcxsacy— 26 (1) 

a(J  — V)     2b3 


be 


a4-2b 

Ans.  x=i-,  v= • 

ic   •  c 


*  These  should  not  bo  cleared  of  fractions,  bat  tbo  unknown  fractions  bo  climi 

Bated  by  making  Uicia  Blike,  and  subtracting. 


y— -r-=& W 


SIMPLE  EQUATIONS.  153 

a             b  ,1X 

:42)  Given  — — =— — - 11) 

ax+2by=d    .  .  .  .  : (2) 

fZ_Ga2+262         3a2+</-6* 
Ans.  z= ^ ,  2/=        T7£- 

(43)  (iiven  x— •— 7—  =  c V1) 

a — x 

6~ 

a_a6+62c+6d         g+<z&— 6c+6art 
Ans.  xz= ^p  ,  y—~       ¥+ !       ""• 

a-4-46     2a— 36  ,-v 

(44)  Given— L. —  =t I1) 

v     '  ?ft+a:      3m, — 2/ 

5ax—2byz=c (2) 

#(c3— 63)     263       ,  m 

(45)  Given  ^+^-^-T=c3.r (1) 

b(cx+2)=cy i (2) 

Ans.  a;=^.  2/  — — ~ 

336 

(46)  Given  17a;— m+(6+10/)2/=/3;c ^ 

4.^+52/=^^ (2) 

a     b  /n 

(47)  Given  -+-=?» I1) 


2/ 
-C+^=n (2) 


6c — ad  be — ad 

Ans'  x=nb~^d:  2/=^=^* 

,48)  Given  x -\-y  =s (!) 

x^—y^—d (2) 

s*+d         s2— </ 
Ans.  .r=-^-,  ys=-jg-. 

(49)  Givena.-+2/:a::a;— 2/:6 (1) 

z2_2,2=C <2> 

a-4-6    Jc  "  —  6    /c 

Ans.  x=-¥-yJ~b,  y=-2-yJZb' 

(50)  Given  .r+  y^-f  7y=a (1) 

x+  ■y/x*—yz=zb (8) 

a3+62  a6(a— 6) 

Ans-  -r=5^+6j'  ^=    a+6 

151)  Given  x2+a:2/=« (*) 

2/24-.ri/=:6 (2) 

a  6 

Ans.  x= —  ,  y= —  ■ 

■y/a  +  6  "       V a-\-b 


154 


ALGEBRA. 


(52)  Given  2:r+3y+4z  =  16 

3x+2y— 5z=  8 
5x—6y+3z=  5 

(53)  Given  5x—Gj-\-iz  =  15. 

7x+4y— 3z=19, 
2ar+  2/  +  Gz=46, 


(1) 
(2) 
(3) 


Ans.  x=3;  2/  =  2;  r  =  l. 


(1) 
(3) 


(54)  Given*  -+- =a 
x     y 

X  '  z 

1    1 

-+-=c 
y  '  z 


Ans.  z=3;  2/=4  !  ~  =  t>. 
.  ..(1) 


Ans.  xz=- 


(2)    ■ 

(3) 

o 


;  2/= 


rt-|-6 — c  a  —  b-\-c  b-i-c — a 

(55)  Given  x+y=36  ;  1+2=49;  7/-fz=53. 

Ans.  z=16;  y=20;  z=33 

(56)  Given  r+ttf+z =30;  »+«?— z=18;  f— ?r+:  =  14. 

Ans.  r=16;  ?r=8;  z=6 

(57)  Given  u+Jt) =164;  t>+fw=82;  7/  +  ]w  =  136. 

Ans.  u=128;  t>=72;  10=40. 


(58)  Given  ax-\-by=c;  my-\-nz=j)  ;  fx-\-gz=zq. 


Ans.  x= 


bnq-\-cgm — b<zp 
agm-\-bfn       ' 
agp-\-cfn — anq 
-*~      agm-\-bfn       ' 
•\-hfp — cfm 


agm  +  bfn 


(C9)  Given  3(ax+by)=z  ;  5y=7(x+3a) ;  llarssfz-f  121. 

o  +  189a& 

Ans.  x  = 


440— 45a  — 636' 
6776+1848a  —  189a" 
y_       440— 45a— 636      ' 
_14520«4-5544<zt  +  203286 
440— 45a— 63i         ' 

7        5     9       11        13         15 
(60)  Given  —=-;  j-j^S  7=yZl3' 

Ans.  r=— 10-;:;  y=-34  ■..  :    :  =  -32. 

a_|_Z»      /; — c      />-}-''      r — </     i/-\-J:     lc — h 
'61)  Giveu — : — =  ; ;  r-. —  = ;  -r-: —  =7 . 

'  a-\-x      b  —  y      b-\-y      r  —  z      </-4-;       l;—x 


*  Do  net  clear  1  is,  but  mako  --,     .  4.C.,  the  unknown  quantities. 


SIMPLE  EQUATIONS. 


155 


(62)  Given  2.i- 


3y  1       1 


7:c—5z=y  -\-x  - 
x     v       z 
3+1  +4  =58 


•  86 


(1) 

(2) 

(3) 

Ans.  £=48;   v=54  ;  2=64. 


(63)  Given    6x— 4y+5z=2fi 

Ax-\-2>y  —  7z  =  l\  . 
12x—Gy  — 32=3  j  . 


(1) 
(2) 
(3) 


164)  Given  18x—7y—5z  =  11 . 
4y-fc+a§*=108 


(65)  Given  y+~+5 


Ans.  x=l;  y  =  \;  z=J. 

...  (1) 
.  .  .  .  (2) 
,  .  .  .  (3) 

Ans.  .r=12;  i/=25;  z=6. 


3     o 

x—1     y—2     2  +  3 
~ 4  5~    :  10 


fy— 5 


-l3  — - 
"   ■»      12 


(66)  Given  ^  +  |  +  ~  58 


5.r      y      z 
7  +  6  +  3 


76 


X        32        u 

2+-8-+5=79 


(1) 

(2) 

(3) 
=5 

(1) 
(2) 

(3) 


y  =  7;  is. 


(4) 

:30;  2=168;  w=50. 


^  +  z  +  M  =248 

Ans.  .r==12  ;  2/= 

(67)  Given  7x— 2z  +  3u=17 (1) ' 

4y—2z+    l  =  \\ (2) 

5y— 3a:— 2«ss  8 (3) 

4y—3u+2t—  9 (4) 

3z  +  8w  =  33 (5) 

Ans.  £=2;  ys=4;  z  =  3;  u=3 


(  =  1. 


Elimination  may  be  effected  in  a  general  form,  and  particular  cases  be  re- 
solved  by  substitution  in  this  form. 

We  shall  illustrate  this  with  a  system  of  three  equations. 

Given  ax  -\-by  +cz   +^  =°» 

a'x  +  b'y  +c'z  +k'  =0, 
a"x+b"y  +c"z+k"=0. 

Eliminating  among  these  three  equations  by  any  of  the  foregoing  methods 
v?e  find 

(b"c'  —b'c")k  +  (be"  —b"c)lc'  +  (b'c  —bc')k" 


.Ts= 


(a'b"—a"b')c+{a"b—ab")c'  +  (ab'  —a'b)c'" 


15G  ALGEBRA. 

(a'c"—  a"c')k  +  {a"c—ac")k'  +  (ac'  —a'c)k" 
J       The  same  denominator  as  in  the  value  of  x   ' 

(a"b'—a'b")k-{-  {ab"—a"b)k'  -f  {a'b — ab')k" 
^        The  same  denominator  as  in  the  value  of  x 
To  apply  this  general  form  to  a  particular  case,  take  (Example  53)  above. 

=  (1X—  3— 4X6)(— 15)  +  (— OXC— 1X4)(— 10j  +  [-»X4—  (— 6X-J3)](— 46)_lg57 
C  (7X1— 2X4)4+(2X— 6— 5X1)(— 3)+(5X4— 7X— I  —  419        ' 

(42+C)(  — 15)  +  (8-30)(  — 19)  +  (  — 15— 28)(— 4G)  _1C7C 
y~  419  ~       -lliJ  ~4' 

(l)(-15)  +  (  +  17)(-19)  +  (-C2)(-46)      2514 
Z—  419  —  419 

Changing  the  signs  of  k,  k',  k",  in  order  that  they  may  be  positive  in  tne 
second  member  of  the  three  proposed  equations,  and  performing  the  multipli- 
cations indicated  in  the  general  values  of  x,  y,  and  z,  they  may  be  written  as 
follows  : 

kb'c"  —kc'b"  +ck'b"  —  bk'c"  +  bc'k"  —  cb'k" 

X=ab'c"  —ac'b"  +  ca'b"  —ba'c"  +  bc'a"  —  cb'a'" 

ak'c"  —ac'k"  +ca'k"  —ka'c"+kc'a"  —ck'a" 

"  The  same  denominator  as  that  of  x  ' 

ab'k"  —  ak'b"  -f  ka'b"  —  ba'k"  -f  bk'a"—kb'a" 

The  same  denominator  as  before 

By  observing  carefully  the  composition  of  the  formulas  for  two  and  three 
equations,  we  may  discover  general  rules  by  means  of  which  wo  can  calcu- 
late the  formulas  suitable  for  any  number  of  equations. 

First  Rule. — To  find  the  common  denominator  in  the  values  of  all  the 
unknown  quantities.  With  the  two  letters  a  and  b  form  the  arrangements 
ab  and  ba,  then  interpose  the  sign  —  between  them,  thus  : 

ab — ba. 

If  there  are  but  two  equations  to  resolve,  placo  an  accent  on  the  2°  letter 
of  each  term,  and  the  result,  ab' — ba',  will  be  the  common  denominator  of 
the  values  of  x  aud  y. 

If  there  are  three  equations,  pass  the  letter  c  through  all  the  places  in  each 
term  of  the  expression  ab — ba,  taking  caro  to  alternate  the  signs  ;  ab  will  thus 
givo  abc — acl-\-cab  ;  also,  — ba  will  give  — bac-\-bca — cba,  and  tho  whole 

abc — acb-\-cab  —  bac-\-bca — cba  ; 
then  place  one  accent  on  the  2°  letter  of  each  term,  and  two  on  the  3 ',  and  the 
resulting  expression  will  be  tho  common  denominator  of  the  values  of.r,  y,  and  z. 

If  there  are  four  equations,  take  the  letter  </,  which  is  the  coefficient  of  the 
fourth  unknown  u,  and  pass  it  through  all  the  plans  in  each  term  of  tho  sexi- 
nomial  above  formed,  taking  caro  to  alternate  the  si^ns  of  the  terms  furnished 
by  each  of  them,  beginning  with  -f-  for  thoso  which  result  from  a  term  pro- 
ceded  by  the  sign  -f-,  and  with  —  for  those  resulting  from  a  term  affected 
with  tho  sign  —  ;  finally,  place  ono  accent  mi  the  2°  letter,  two  on  the  3",  and 
three  on  the  4".  Tho  resulting  polynomial  is  the  common  denominator  of  the 
four  unknown  quantities  x,  y.  z,  u. 

ab'cf'd'  '  —  b'd"d"+ad'h"t?"— da'b"&" 

—ac'b"</'"  +  ,!y,/,h"—l„lr'ir+l/„'c"b'" 
±ca,l>">i—ca'd"b"'  +  cd'a"b'   —  b'" 


SIMPLE  EQUATIONS.  157 

—  ba'c"d"' + ba'd"c'"  —  bd'a"c'"-\- db'c  c'" 
+  bc'a"d"'—bc'd"a"'+bd'c"a'"—db'c"a" 
—cb'a"d'" -\-cb'd"a'" —cd'b"a"' +dc'b" a' ' . 

if  there  lie  a  greater  Dumber  of  equations,  proceed  in  the  same  manner. 

Second  Rule. — The  numerators  may  be  derived  from  the  common  de 
nominator.     For  this  purpose,  it  is  only  necessaiy  to  replace,  without  touch 
ing  the  accents,  tho  letter  which  serves  for  coefficient  of,  the  unknown  quanta 
ty  we  wish  to  find,  by  the  letter  k,  which  represents  tho  known  term  in  the 
second  member.     Thus  :  change  a  into  k,  to  have  the  numerator  of  x ;  b  into 
k,  to  have  that  of  y  ;  and  so  on. 

There  remains  still  a  method  of  elimination  to  bo  mentioned,  which  alone 
is  applicable  to  equations  of  higher  degrees,  as  well  as  to  those  of  the  first.  It 
u  called  the  method  of  tho  common  divisor.  It  consists,  where  two  equations 
are  given,  in  dividing  one  by  the  other  (after  transferring  all  the  terras  to  the 
first  member  in  both),  that  divisor  by  the  remainder,  and  so  on  till  tho  letter 
jf  arrangement,  which  must  be  one  of  the  unknown  quantities,  is  exhausted 
from  the  remainders.  The  last  remainder  containing  but  the  other  unknown 
quantity,  being  put  equal  to  zero,  will  present  an  equation  from  which  the  first 
unknown  quantity  is  eliminated. 

If  there  bo  three  or  more  equations,  eliminate  one  of  tho  unknown  quanti- 
ties in  this  way  between  the  first  and  second,  then  between  the  first  and  third, 
and  so  on. 

Tho  reason  which  may  be  given  for  this  rule  here,  though  a  better  one  win 
be  furnished  hereafter,  is,  that  the  dividend  being  zero  and  the  divisor  zero, 
the  remainder  must  be  zero. 

Let  us  apply  this  method  to  Example  (8)  above.    The  two  g'ven  equations  are 

X+   y— 10  =  0 
2x—3y—  5=0. 

Elimination, 


2x—3y  —  5 
2x4-2^—20 


z+y-io 


-57/+15  -4-5. 
-  t/4-  3  =0  .-.  y=3. 
Substituting  this  value  in  x-\-y — 10  =  0,  we  obtain  z=7. 

EXAMPLE   II. 

Given.r-34-3iy.i-4-3/.r— 98=0 (1) 

3?-\-4yx  —2y~   —10=0. 
Elimination, 


x*-\-2yxi-\-  2y2x—    98 
x*-\-iyx*—  Zyix—    10.r 


x--{-4yx — 2^2 — io 


x—y 


—  yx--{-  5y*x-\-  lOx — 98 

—  yx- —  4//2.r-j-  2,7/3-f-10# 
x*-\-   Ayx—    Zy? — 10 

9//2+10 


9y*x-\-  10  x—  2?/3—  Wy— 98  or 
{9y*  -j-10)  x—  2?/3— 10?/— 98 


(9^4-10):^-f(36y3-J_40y)j>_i8y«— 110^2— 100|a;4-]ty34-25$r+49 

(9yg+10)a^— (  2y3-f.iQy   4.98)3; 

(WyZ+SQy    -{-98);e—  18y*— HOyS— 100---2 
(19yS-j-25y   4-49);r—     9yi—  55^3—  50 

. 9yH-  10 

(9ys+10)(19y3_L.o5^  +49)2:—  81y6—585y*— 1000^—  500 
(9^2+10)  (19  ;/*-f25y   -|-49).r —  38?/"— 2 40y*— 1060?/3— 250?/?-  .2P40?/—  4 801! 

—  43^i— 345^-«4-1960^3— 750j/--j-2!)40?/4-4e0? 


153  ALGEBRA. 

This  last  remainder,  put  equal  to  zero,  will  make  an  equation  from  whiih  x 
is  eliminated,  and  which  contains  only  y.     It  is  called  the  final  equation. 

ON  THE  SOLUTION  OF  PROBLEMS  WHICH  PRODUCE  SIMPLE 

EQUATIONS. 

150.  Every  problem  which  can  be  solved  by  Algebra  includes  in  its  enun- 
ciation a  certain  number  of  conditions  of  such  a  kind  that,  in  taking  at  pleasure 
values  for  the  unknown  quantities,  it  is  always  easy  to  see  whether  or  not  they 
will  verify  these  conditions.  In  the  greater  part  of  questions  in  Algebra,  these 
verifications  consist  in  this,  that,  after  having  effected  certain  operations  upon 
the  values  of  the  known  and  unknown  quantities,  we  ought  to  arrive  at  equali- 
ties. This  being  understood,  if  the  unknown  quantities  bo  represented  by 
letters,  algebraic  expressions  may  be  formed  in  which  shall  be  indicated,  by 
means  of  signs,  all  tfee  calculations  necessary  to  be  made,  as  well  upon  the  un- 
known numbers  as  upon  the  known,  to  find  the  quantities  which  ought  to  be 
equal.  Consequently,  joining  these  expressions  by  the  sign  of  equality,  we 
shall  havo  one  or  more  equations,  which  will  be  satisfied  when  the  true  val- 
ues of  the  unknown  quantities  are  substituted  in  the  place  of  the  letters  which 
represent  them. 

Reciprocally,  when  all  the  conditions  of  the  problem  are  expressed  in  the 
equations,  the  values  of  the  unknown  quantities  which  satisfy  these  equations 
must  certainly  satisfy  the  enunciation  of  the  problem. 

It  is  impossible  to  give  a  general  rule  which  will  enable  us  to  translate  eve- 
ry problem  into  algebraic  language  ;  this  is  an  art  which  can  bo  acquired  by 
reflection  and  practice  alone.  Two  rules  which  may  be  of  some  service  are 
the  following  :  1.  Indicate  upon  the  unknown  quantities  represented  by  letters, 
and  upon  the  known  quantities  represented  either  by  letters  or  numbers,  the  same 
operations  as  uvuld  be  necessary  to  verify  them  if  they  xcere  known.  2.  Form 
two  different  expressions  of  the  same  quantity,  and  set  them  equal.  We  shall 
give  a  few  examples,  which  will  serve  to  initiate  the  6tudent,  and  the  rest 
must  be  left  to  his  own  ingenuity. 

PROBLEM  1. 

To  find  two  numbers  such  that  their  sum  shall  be  40,  and  their  difference 
16. 
Let  x  denote  the  least  of  the  two  numbers  required, 
Then  will  .r-f-16=  the  greater, 

And  x-\-x -f-16  =  40  by  the  question  ; 

That  is,  &r=  40— 16=24  \ 

Or  x=— =  12  =  less  number, 

And  x-f-16  =  12-j-16=28=  greater  number  required. 


What  number  is  that,  whose  '.  pari  exceeds  its  I  part  by  161 

Let  x=  number  requii 

Then  will  its  \  part  be   \x,  and  its  J  par! 

And,  therefore,  '   — |  ■— n;  by  the  question, 
Or,  clearing  of  fractions,  l.r  —  3x=rl92  : 
Hence  X=192,  the  number  required. 


SIMPLE  EQUATIONS.  159 

PROBLEM  3. 

Divide  c£l000  among  A,  B,  and  C,  so  that  A  shall  ha~e  <£72  more  than  B. 
and  C  <£100  more  than  A. 

Let  *      x=  B's  share  of  the  given  sum, 

Then  will         ar+  72=  A-'s  share, 

And  ar+172=  C's  share, 

And  the  sum  of  all  their  shares,  :r+:r+72+.r+172, 

Or  3x+ 244  =  1000  by  the  question ; 

That  is,  3.t= 1000  — 244  =  756, 

756 
Or  =—r=de252=  B's  share  ; 

Hence  x+  72=252+  72=,€324=  A's  share, 

And  .r+172  =  252+172=c£42t=  C's  share; 

B's  share o€252 

A's  share 324 

C's  share 424 

Sum  of  all  .  .  <£1000,  the  proof. 

problem  4. 

Out  of  a  cask  of  wine,  which  had  leaked  away  -*,  21  gallons  were  drawn, 
and  then,  being  gauged,  it  appeared  to  be  half  full:  how  much  did  it  hold  ? 
Let  it  bo  supposed  to  have  held  x  gallons, 
Then  it  would  have  leaked  *x  gallons  ; 
Consequently,  there  had  been  taken  away  21  +  ]£  gallons. 
But  2 l-f-?a;=ir  by  the  question,  * 

Or  12G  +  2.r=3.r; 

Hence        3.r— 2.r=126, 
Or      £=126=  number  of  gallons  required. 

problem  5. 

A  hare,  pursued  by  a  greyhound,  is  60  of  her  own  leaps  in  advance  of  tH3 
dog.  She  makes  9  leaps  during  the  time  that  the  greyhound  makes  only  6: 
but  3  leaps  of  the  greyhound  are  equivalent  to  7  leaps  of  the  hare.  How 
many  leaps  must  the  greyhound  make  before  he  overtakes  the  hare  ? 

It  is  manifest,  from  the  enunciation  of  the  problem,  that  the  space  which 
must  be  traversed  by  the  greyhound  is  composed  of  the  60  leaps  which  the 
hare  is  in  advance,  together  with  the  space  which  the  hare  passes  over  from 
the  time  that  the  greyhound  starts  in  pursuit  until  he  overtakes  her. 

Let  x=  the  whole  number  of  leaps  made  by  the  greyhound.     Since  the 

hare  makes  9  leaps  during  the  time  that  the  greyhound  makes  6,  it  follows 

9         3 
that  the  hare  will  make  -  or  -  leaps  during  the  time  that  the  greyhound 

3.r 

makes  1,  and  she  will  consequently  make  —  leaps  during  the  time  that  the 

greyhound  makes  x  leaps. 

We  might  here  suppose  that,  in  order  to  obtain  the  equation  required,  it 

3a: 

would  be  sufficient  to  put  x  equal  to  60+—;   in  doing  this,  however,  we 

should  commit  a  manifest  mistake,  for  the  leaps  of  the  greyhound  are  greater 


160  ALGEBRA. 

than  the  leaps  of  the  hare,  and  we  should  thus  be  equating  two  heterogeneous 
numbers;  that  is  to  say,  numbers  related  to  a  different  unit.  In  order  to  re- 
move this  difficulty,  we  must  express  the  leaps  of  the  hare  in  terms  of  the 
leaps  of  the  greyhound,  or  the  contrary. 

According  to  the  conditions  of  the  problem,  3  leaps  of  the  greyhound  are 

7 
equal  to  7  leaps  of  the  hare ;  hence  1  leap  of  the  greyhound  is  equal  to  - 

?.r 
leaps  of  the  hare,  and,  consequently,  x  leaps  of  the  greyhound  are  equal  to  — 

leaps  of  the  hare  ;  hence  wo  have  at  length  the  equation 

7x  3x 

Clearing  of  fractions,  14a:r=360-|-9:r 

x—  7-2. 
Hence  the  greyhound  will  make  72  leaps  before  he  reaches  the  hare,  and  m 

that  time  the  hare  will  make  72  X^  or  108  leaps. 

PROBLEM    6. 

Find  a  number  such,  that  when  it  is  divided  by  3  and  by  4,  and  the  quo 
tients  afterward  added,  the  sum  is  63. 

Let  x  be  the  number  ;  then,  by  tho  conditions  of  the  problem,  we  have 

x     x 
3+4=  C3; 

I  Hearing  of  fractions,  7x=  63X12 

.r=108. 
If  we  wished  to  find  a  number  such  that,  when  divided  by  5  and  by  6,  the 
sum  of  the  quotients  is  22,  we  must  again  translate  the  problem  into  algebraic 
language,  and  then  solve  the  equation  ;  in  this  case  we  have 

x         x 

5+     6 ' 

Clearing  of  fractions  1  l.r =22  X  30 

r=60. 
If,  however,  wo  desire  to  solve  both  these  problems  at  once,  and  all  others 
of  tho  same  class,  which  differ  from  tho  above  in  tho  numerical  values  only, 

we  must  substitute  for  thoso  particular  numbers  the  symbols  a,  b,  c, , 

which  may  represent  any  numbers  whatever,  ami  then  solve  the  following 
question. 

Find  a  number  such  that,  when  it  is  divided  by  a  and  by  b,  and  the  quo- 
tients afterward  added,  tho  sum  is  p.     We  have 

x     x 

-+T     =i>i 

a  '  b         1 

(ii-\-b).v=  ahp 
ahp 

151.  This  expression  is  not,  strictly  speaking,  the  value  of  tho  unknown 
quantity  in  OUT  problems,  but  it  presents  to  OUT  view  the  calculations  which 
an  r<  Min  -Hi-  for  the  solution  of  them  all.      \x  I  MB  of  this  nature  is  call- 


SIMPLE  EQUATIONS.  161 

ed  a  formula.  This  formula  points  out  to  us  that  the  unknown  quantity  is  ob- 
tained by  multiplying  together  the  three  numbers  involved  in  the  question, 
and  then  dividing  their  product,  abp,  by  a-\-b,  the  sum  of  the  two  divisors  ;  or 
we  should  rather  say,  that  our  formula  is  a  concise  method  of  enunciating  tho 
above  rule.*  Algebra,  then,  may  be  considered  as  a  language  whose  object 
«s  to  express  various  processes  of  reasoning,  as  also  the  results  or  conclusions 
to  which  they  lead. 

Such  is  the  advantage  of  the  above  formula,  that,  by  aid  of  it,  the  most  ig 
norant  arithmetician  could  solve  either  of  the  proposed  problems  as  readily  as 
the  most  expert  algebraist.     The  former,  however,  could  only  arrive  at  the 
result  by  a  blind  reliance  on  the  rule  which  the  formula  expresses  ;  but  differ- 
ent kinds  of  problems  require  different  formulae,  and  the  algebraist  alone  pos 
sesses  the  secret  by  which  they  can  be  discovered. 

PROBLEM    7. 

A  laborer  engaged  to  servo  40  days  upon  these  conditions  :  that  for  every 
day  he  worked  he  was  to  receive  80  cents,  but  for  every  day  he  was  idle  he 
was  to  forfeit  32  cents.  Now  at  the  end  of  the  time  he  was  entitled  to  re- 
ceive $15.20.  It  is  required  to  find  how  many  days  he  worked  and  how 
many  he  was  idle. 

Let  x  be  the  number  of  days  he  worked  ; 

Then  will  40 — x  be  the  number  of  days  he  was  idle  ; 

Also  .rx  80  =  80.r=  the  sum  earned, 

And  (40— x)  X  32=1280— 32.r=  sum  forfeited  ; 

Hence  80.r—  (1280— 32.r)  =  1520  by  the  question  ; 

That  is,  80x— 1280  +  32.r=1520, 

Or  112.r=1520  + 1280=2800  ; 

2800 
Hence  .r=Tpr^-=25=  number  of  days  he  worked, 

And  40 — .r=40 — 25  =  15=  number  of  days  ho  was  idle. 
We  may  generalize  the  above  problem  in  the  following  marner  : 
Let  n=  tho  whole  number  of  days  for  which  he  is  hired, 

a=  the  wages  for  each  day  of  work, 
b  =  the  forfeit  for  each  day  of  idleness, 
c=  the  sum  which  he  receives  at  theftnd  of  n  days, 
x=  the  number  of  days  of  work  ; 
Then  n — .t=  the  number  of  days  of  idleness, 

ax=  the  sum  due  to  him  for  the  days  of  work, 
b(n — .r)=  tho  sum  he  forfeits  for  the  days  of  idleness. 
We  thus  find  for  the  equation  of  the  problem, 

ax — b  (?i — x)  =   c  ; 
Whence  ax — bn  -\-bx—  c 

(a-\-b)x=  c-^-bn 

c-\-bn 
x=  — — r-.  the  number  of  days  of  work, 
a+b  J 


*  Let  the  student  try  this  rule  upon  a  variety  of  numbers  ;  he  will  see  that  the  generi 
brmula  embraces  as  many  particular  examples  as  he  chooses  to  imagine. 

L 


162  ALGEBRA. 

e-\-bn 


And  .•.  n — x=  n 


a+b 
an-\-bn — c  —  ha 

a  +  b 
an — c 


,  .    llio  number  of  dt  ,-s  of  idleness. 
a-\-b 

By  substituting  in  these  general  expressions,  for  the  number  of  days  of 

work  and  number  of  days  of  idleness,  the  particular  numerical  values  of  the 

letters,  the  samo  result  will  be  obtained  as  before. 

problem  8. 

A  can  perform  a  piece  of  work  in  G  days,  B  can  perform  the  same  worK  in 
8  days  :  in  what  timo  will  they  finish  it  if  both  work  together  ? 
Let  .r=  the  time  required. 

Since  A  can  perform  the  whole  work  in  G  days,  -  will  denote  the  quantity 

x 
he  can  perform  in  1  day,  and  therefore  -  the  quantity  ho  can  perform  in  .r 

days ;  for  the  same  reason,  -  will  bo  the  quantity  which  B  can  perform  in  x 

days ;  and  we  shall  thus  have 

x     x 

6+8  =  1t 
14.r=48 
x=3!)  days. 
Let  us  generalize  the  above  problem. 

A  can  perform  a  piece  of  work  in  a  days,  B  in  b  days,  C  in  c  days,  D  in  d 
days  :  in  what  time  will  they  perform  it  if  they  all  work  together  ? 
Let  .r=  the  time; 

Then,  since  A  can  perform  the  whole  work  in  a  days,  -  will  denote  the 

x 

quautity  he  can  perform  in  1  day,  and,  consequently,  -  will  be  the  quantity  he 

x  x  x 
can  perform  in  x  days;  for  the  samo  reason,  -r,  -,  —.  will  be  the  quantities 

which  B,  C,  D  can  perform  respectively  in  x  days  ;  we  thus  have 
^+i+^=(wll0leW01'k)' 

=i; 

abed 
abc-\-abd-\-acd-\-bcd" 
What  is  the  rulo  expressed  by  this  formula  ? 


*  Let  tbo  student  translate  the  formula  for  the  number  of  days  of  idleness,  and  that  for 
the  number  of  days  of  work,  into  a  rule. 

V        P 

t  We  might  represent  the  piece  of  work  by  p ;  then  ,  and  -  would  express  the  quantities 

which  A  and  B  can  perfoua  in  ono  day,  and  tlio  equation  would  be 

i,  divided  throughout  bpp,  gives  the  equation  in  the  text    WLenthc  valao  of  aquaa 
titv  in  immaterial,  as  in  this  case,  it  is  best  represented  by  1. 


SIMPLE  EQUATIONS.  163 

PROBLEM  9. 

A  courier,  who  traveled  at  the  rate  of  31  ^  miles  in  5  hours,  was  dispatched 
from  a  certain  city ;  8  hours  after  his  departure,  another  courier  was  sent  to 
overtake  him.  The  second  courier  traveled  at  the  rate  of  22^  miles  in  3  hours. 
In  what  time  did  he  overtake  the  first,  and  at  what  distance  from  the  place  of 
departure  ? 

Let  x=  number  of  horns  that  the  second  courier  travels. 

Then,  since  the  first  courier  travels  at  the  rate  of  31  J-  miles  in  5  hours,  that 

pi)  />Q 

is,  r-7.  miles  in  1  hour,  ho  will  travel  —  x  miles  in  x  hours,  and,  since  he  start 
ed  8  hours  before  the  second  courier,  the  whole  distance  traveled  by  him  will 
be  (S+x)-. 

Again,  since  the  second  courier  travels  at  the  rate  of  22^  miles  in  3  hours 

45  45        .       . 

that  is,  —  miles  in  one  hour,  ho  will  hence  travel  —  x  miles  in  x  hours. 

The  couriers  are  supposed  to  be  together  at  the  end  of  the  time  x,  and 
therefore  the  distanco  traveled  by  each  must  be  the  same ;  hence 

45  63 

-*=(8+*)- 

450.r=(8+.r)378; 

.-.  72.r=3024 

a: =42. 

Hence  the  second  courier  will  overtake  the  first  in  42  hours,  and  the  whole 

45 
distance  traveled  by  each  is  —  X  42=315  miles. 

To  generalize  the  above, 

A  B  C 


!  1  I 

Let  a  courier,  who  travels  at  the  rate  of  m  miles  in  t  hours,  be  dispatched 
from  B  in  the  direction  C ;  and  n  hours  after  his  departure,  let  a  second 
courier,  who  travels  at  the  rate  of  m'  miles  in  V  hours,  be  sent  from  A,  which 
is  distant  a.  miles  from  B,  in  order  to  overtake  the  first.  In  what  timo  will  he 
come  up  with  him,  and  what  will  be  the  whole  distance  traveled  by  each  ? 

Let  x—  number  of  hours  that  the  second  courier  travels. 

Then,  since  the  first  courier  travels  at  the  rate  of  m  miles  in  t  hours,  that  is, 

7th  7Tb 

—  miles  in  1  hour,  he  will  travel  —x  miles  in  .r  horns,  and,  since  he  started  n 
t  c 

hours  before  the  second  courier,  the  whole  distance  traveled  by  him  will  be 

—    m 
(n+x)-. 

Again,  since  the  second  courier  travels  at  the  rate  of  m'  miles  in  V  hours, 

7th  771 ' 

that  is,  —  miles  in  1  hour,  he  will  travel  —  x  miles  in  x  hours ;  but  since  he 

started  from  A,  which  is  distant  d  miles  from  B,  the  whole  distance  traveled 

m' 
by  the  second  courier,  or  —x,  will  be  greater  than  the  wt.ole  distance  traveled 

by  the  first  courier,  by  this  quantity  d ;  hence 

\ 


164  ALGEBRA. 


m'  m 

—  x—d  =  {n  +  x)- 


t'  *     »     >t 

(in'     m\        mn 

(mn+fo£)*/ 


m7  —  mt' 


1  he  wnuie  distance  traveled  by  first  courier,        =—  .  { ■ 


m't — mt' 


The  whole  distance  traveled  by  second  courier,  =— 


f       m't  —  mf 


PROBLEM  10. 


A  father,  who  has  three  children,  bequeaths  his  property  by  will  in  the  fol- 
lowing manner :  To  the  eldest  son  he  '  sum,  a,  together  with  the  /;'"  part 
of  what  remains  ;  to  the  second  he  leaves  a  sum,  2a,  together  with  the  n' 
of  what  remains  after  the  portion  of  the  eldest  and  2a  have  been  subtracted ; 
to  the  third  ho  leaves  a  sum,  3a,  together  with  the  nth  part  of  what  remains 
after  the  portions  of  the  two  other  sons  and  3a  have  been  subtracted.  The 
property  is  found  to  be  entiroly  disposed  of  by  this  arrangement.  Required 
the  amount  of  the  property. 

Let  x=  the  property  of  the  father. 

If  we  can,  by  means  of  this  quantity,  find  algebraic  expressions  for  the  por- 
tions of  the  three  sons,  we  must  subtract  their  sums  from  the  whole  pro) 
x,  and,  putting  this  remainder  =0,  we  shall  determine  the  equation  of  the 
problem. 

Let  us  endeavor  to  discover  these  three  portions. 

Since  x  represents  the  whole  property  of  the  father,  x — a  is  the  remainder 
after  subtracting  a  ;  hence, 

x — a 
Portion  of  eldest  sou,    =a-\- 


n 

an-\-x— 


n 

an-\-x — a 
x — 2a  — 


(1) 


71 

Portion  of  second  son,  =2a-f- 


=2a4 


n 

nx — 'Ian  —  r-\-(i 


•!uir-\-nx — 3an  — r+a 


x— 3a 


aii-\-.r — a  '-nx — 3an — x-\-a 


(2) 


Portion  of  third  BOO,       =  3</-^ 


=  3a-l 


n 


3gn'+n«r— 6an*— 2nx+4an+z— a 
According  to  the  conditions  of  the  problem,  the  pro]  itirerjrdisp 


SIMPLE  EQUATIONS.  165 

rf.     Hence,  when  the  sum  of  the  three  portions  is  subtracted  from  a,  the  dif- 
ference must  be  equal  to  zero  ;  this  gives  us  the  equation 

an-\-x — a      larP-^-nx — 3<zm — x-\-a      "iaip-^rflx — Qan2 — 2na;-\-ian-\-x — a 

clearing  the  equation  of  fractions,  and  reducing, 

,—6arii—3ri2x-\-10an'i-{-3nx—5an—x-\-a=0 
...  {n*—3n*+3n— l)x=6a?i3— 10are2-|-5an  —  a 

6an?— 10a7i2+5aw— a     (6w3— 10n--\-5n  —  l)a 
X~      ?i3— 3?i-+3/i— 1      =  {n— 1)»  ' 

reflecting  upon  the  conditions  of  the  problem,  we  may  obtain  an  equation 
much  more  simple  than  the  preceding.  It  is  stated  that  the  portion  of  tho 
third  son  is  3a,  together  with  tho  »th  of  what  remains,  and  that  the  property 
is  thus  entirely  disposed  of  ;  in  other  words,  the  portion  of  the  third  son  is  3a, 
and  the  remainder  just  mentioned  is  nothing. 

We  found  the  expression  for  that  remainder*  to  be 
rPx — 6an2 — 2nx-\-4an-\-x — a 

Equating  this  quantity  to  zero,  we  have 

.•.  n2x — 6an" — 2nx-{-Aan-\-x — a  =  0 
(n° — 2n-\-l)x=6ari2 — Aan-\-a 
Gan- — 4ara-f-a 

»a — 2n -j- 1 
(6n-— 4n  +  l)a 


.r= 


-       (n-1)"       * 

This  result  is,  moreover,  more  simple  than  the  former.  We  can  easily  prove 
that  the  two  expressions  are  numerically  identical,  for,  applying  to  the  two 
polynomials  (G«3  —  10/t-  +  5>i  —  l)a,  and  (n3 — 3n-+3>i-\-l),  the  process  for  find- 
ing the  greatest  common  measure,  we  shall  find  that  these  two  expressions 
have  a  common  factor  n — 1 ;  dividing,  therefore,  both  terms  of  the  first  result 
by  this  common  factor,  we  arrive  at  the  second. 

The  above  problem  will  point  out  to  the  student  the  importance  of  examin- 
ing with  great  attention  the  enunciation  of  any  proposed  question,  in  order  to 
discover  those  circumstances  which  may  tend  to  facilitate  the  solution ;  he  will 
otherwise  run  the  risk  of  arriving  at  results  more  complicated  than  the  nature 
of  the  case  demands. 

The  above  problem  admits  of  a  solution  less  direct,  but  more  simple  and 
elegant  than  those  already  given.  It  is  founded  on  the  observation  that,  after 
having  subtracted  3a  from  the  former  portions,  nothing  ought  to  remain. 

Let  us  represent  by  r,,  r2,  r3  the  three  remainders  mentioned  in  the  enun 
ciation ;  the  algebraic  expressions  for  the  three  portions  must  be 

r,  r„  r3 

"+*  2a+t'  3a+ ~n 

1".  By  the  conditions  of  the  problem,  we  have  r3=0. 

Hence  the  third  portion  is  3a. 

*  Next  above  (3). 


166  ALGEBRA. 


r, 


2°.  Tho  remainder,  after  the  second  son  has  received  Ba-f  — ,  may  be  rep- 


n 


r„         (n  —  l)r 
resented  by  r„ =,  or  - 


n  n 

But  this  is  the  portion  of  the  third  son  ;  hence  we  have 

(n — l)r„ 

• — =3a 

n 

San 


n — 1 

tt  .         „    ,  ,  San  3a 

Hence   tho   portion  of  the  second  son  is  2a  + r— n  =  2a-f- r»  or» 

1  '  n —  1  '  n  —  1 

reducing, 

2an-{-a 

~n~=T' 

T 

3°.  The  remainder,  after  the  eldest  son  has  received  a-\-—i  may  be  repre- 

eented  by  r, ,  or — -. 

J  n  n 

But  this  remainder  forms  the  portion  of  the  other  two  sons  ;  hence  we  have 

(n  —  l)r,      2an-{-a 


n  n — 1 

ban* — 2an 


3a 


■'■  r'~   {n-iy 

,,  ,  »   ,  ban2 — 2an  ban  —  2a 

Hence  the  portion  of  the  eldest  son  is  a+— ; -rz—  -±-n=a+-, ttt, 

1      (n  —  l)2  '    (n  —  1)' 

or,  reducing, 

an^-^San — a 

1  n-— 2m  +  1 

Hence  the  whole  property  is 

2ara+a     a«2+3an — a 
3a  4- —  4- — ; 

reducing  tho  whole  to  a  common  denominator, 

3a{n0-—2n  +  l)  +  {2,i n  +  ■/)(/<  —  l)4-an24-3ara— Jt  , 

LI 

performing  the  operations  indicated,  and  reducing, 

(6/i8— 4«4-l)a 
n»— 2«4-l    ' 
the  result  obtained  above. 

This  solution  is  more  complete  than  the  former,  fur  we  obtain  at  the  same 
time  the  property  of  the  father  and  the  expressions  for  the  portions  of  his 
three  sons. 

We  shall  now  solve  one  or  two  problems  in  which  it  is  either  I  y  oi 

convenient  to  employ  more  than  one  unknown  quantity. 

problem   1 1. 

Required  two  numbers  whose  sum  is  70  and  whose  difference  is  16 
Let  x  and  y  l><;  the  two  numbers. 
Then,  by  the  conditions  of  the  problem, 


SIMPLE  EQUATIONS.  167 

.r+2/=70 (1) 

x— y  =  l6 (2), 

which  aro  tho  two  equations  required  for  its  solution. 
Adding  the  two  equations, 

2or=86 
x=43. 
Subtracting  tho  second  from  tho  first, 

2y=5i 
y=27. 
Hence  43  and  27  are  the  two  numbers. 

problem  12. 
on  has  two  kinds  of  gold  coin,  7  of  the  larger,  together  with  12  of  the 
smaller,  make  238  shillings ;  and  12  of  the  larger,  together  with  7  of  the  smaller, 
make  358  shillings.     Required  the  value  of  each  kind  of  coin. 

Let  x  be  tho  value  of  the  larger  coin  expressed  in  shillings,  y  that  of  the 
smaller. 

Then,  by  the  conditions  of  the  problem, 

7a?-|r-12i/=288 (1) 

And 

12;r+  7#=358 (2). 

Multiplying  equation  (1)  by  7.  and  equation  (2)  by  12, 
and  subtracting  tho  former  product  from  the  latter,      .     .  95.r=2280 

.-.  x=     24. 
Substituting  this  value  of  x  in  equation  (1),  it  becomes    168  +  12?/=  283 

.-.  y=     10. 
The  larger  of  the  two  coins  is  worth  24  shillings,  the  smaller  10  shillings. 

PROBLEM   13. 

An  individual  possesses  a  capital  of  §30,000,  for  which  he  receives  interest 
at  a  certain  rate  ;  he  owes,  however,  §20,000,  for  which  he  pays  interest  at  a 
certain  rate.  The  interest  he  receives  exceeds  that  which  he  pays  by  §800. 
Another  individual  possesses  a  capital  of  §35,000,  for  which  he  receives  inter- 
est at  tho  second  of  the  above  rates ;  ho  owes,  however,  §24,000,  for  which 
he  pays  interest  at  the  first  of  the  above  rates.  The  interest  which  he  re- 
ceives exceeds  that  which  he  pays  by  §310.  Required  the  two  rates  of  in- 
terest. 

Let  x  and  y  denote  the  two  rates  of  interest  for  §100. 

In  order  to  find  the  interest  of  §30,000  at  tho  rate  .t,  we  have  the  pro 

portion, 

30,000a: 
100: 30,000  ::x:  =300x. 

In  like  manner,  to  find  the  interest  of  §20,000  at  the  rate  of  y, 

20,000'/ 
100 :  20,000 ::  y :— — -^=200?/. 

But,  by  the  enunciation  of  tho  problem,  the  difference  of  these  two  sums  is 
$800  ;  hence  we  shall  have,  for  the  first  equation, 

300.r— 200j/=800 (1). 

Translating,  in  like  manner,  the  second  condition  of  the  problem  into  alge- 
braic language,  we  arrive  at  the  second  equation, 


168  ALGEBRA. 

350y— 240.r=310 (2) 

The  two  members  of  the  first  equation  are  divisible  by  100,  and  those  of  the 
second  by  10  ;  they  may  therefore  be  replaced  by  the  following  : 

3x—  2y=   8 (3) 

35y— 24ar=31 (4) 

In  oi'Jer  to  eliminate  x,  multiply  equation  (3)  by  8,  and  then  add  equation 
(4) ;  hence 

19y=95 

■••  y=  s. 

Substituting  this  value  of  y  in  equation  (3),  we  have 

3.r— 10=8 
.-.  x—6. 
Then  the  first  rate  of  interest  is  6  per  cent.,  and  the  second  5  per  cent. 

problem  14. 

An  artisan  has  three  ingots  composed  of  different  metals  melted  together. 
A  pound  of  the  first  contains  7  oz.  of  silver,  3  oz.  of  copper,  and  6  oz.  of  tin. 
A  pound  of  the  second  contains  12  oz.  of  silver,  3  oz.  of  copper,  and  1  oz.  of 
tin.  A  pound  of  the  third  contains  4  oz.  of  silver,  7  oz.  of  copper,  and  5  oz. 
of  tin.  How  much  of  each  of  these  three  ingots  must  he  take  in  order  to 
form  a  fourth,  each  pound  of  which  shall  contain  8  oz.  of  silver,  3J  oz.  of  cop- 
per, and  4  j  oz.  of  tin  ? 

Let  x,  y,  and  z  be  the  number  of  ounces  which  he  must  take  in  each  of  the 
ingots  respectively,  in  order  to  form  a  pound  of  the  ingot  required. 

Since,  in  the  first  ingot,  there  are  7  oz.  of  silver  in  a  pound  of  1G  oz.,  it  fol- 

7 
lows  that  in  1  oz.  of  the  ingot  there  are  —  oz.  of  silver,  and,  consequently,  in  i 

7x 
oz.  of  the  ingot  there  must  be  —  oz.  of  silver.     In  like  manner,  we  shall  find 

lo 

,       12?/   4z 

that  — '-,  —  represent  the  number  of  ounces  of  silver  taken  in  the  second  and 
10    lb 

third  ingots  in  order  to  form  the  fourth  ;  but,  by  the  conditions  of  the  prob 

lem,  the  fourth  ingot  is  to  contain  8  oz.  of  silver  ;  wo  shall  thus  have 

7x     12?/      42 

ig+Tu-+h]=s W 

And  reasoning  precisely  in  the  samo  manner  fur  the  copper  and  tin,  we  find 
3a:      3?/      7r_15 

16+  16  +  16~~T ™ 

6x       y       5z      17 

16+  16  +16=T (3) 

which  are  the  three  equations  required  for  the  solution  of  tho  problem 
Clearing  them  of  fractions,  they  become 

7s+12y+4z=128 (1) 

3x+  3y+7z=  60 (5) 

6ar-f.     i/+5:=  68 (6) 

In  these  three  equations  the  coefficients  ofy  are  most  simple  ;  it  will,  ther» 
fore,  be  convenient  to  eliminate  this  unknown  quantity  first. 


SIMPLE  EQUATIONS.  1G!) 

Multiply  equation  (5)  by  4,  and  subtract  equa- 
tion (4)  from  the  product,  wo  have 5;r-f-24z=112  .  .  (7) 

Multiply  equation  (6)  by  3,  and  subtract  equa- 
tion (5)  from  the  product,  we  have 15.r-|-  8z  =  144  .  .  (8) 

Multiply  equation  (8)  by  3,  and  subtract  equa- 
tion (7)  from  the  product,  we  have 40.r=320 

.-.  x=     8 

Substitute  this  value  of  x  in  equation  (8) ;  it  be- 
comes          120-f-  8z=144 

.-.  z=     3 

Substitute  these  values  of  x  and  z  in  equation 
(6) ;  it  becomes 48+3/+15  =   68 

•••  V=    5 

Hence,  in  order  to  form  a  pouud  of  the  fourth  ingot,  he  must  take  8  ounces 
of  the  first,  5  ounces  of  the  second,  and  3  ounces  of  the  third. 

problem  15. 

There  are  threo  workmen,  A,  B,  C.     A  and  B  together  can  perform  a  cer- 
tain piece  of  labor  in  a  days ;  A  and  C  together  in  b  days ;  and  B  and  C  to- 
gether in  c  days.     In  what  time  could  each,  singly,  execute  it,  and  in  what 
time  could  they  finish  it  if  all  worked  together  ? 
Let  x==  time  in  which  A  alone  could  complete  it. 
y=  time  in  which  B  alone  could  complete  it. 
z  =  time  in  which  C  alone  could  complete  it. 
Since  A  and  B  together  can  execute  the  whole  in  a  days,  the  quantity 

which  they  perform  in  one  day  is  -  ;  and  since  A  alone  could  do  the  whole 

in  x  days,  the  quantity  he  could  perform  in  one  day  is  - ;  for  the  same  rea 

son,  the  quantity  which  B  could  perform  in  one  day  is  - ;  the  sum  of  what 

D 
they  could  do  singly  must  bo  equal  to  the  quantity  they  can  do  together 

hence  / 

111 

-+-=- 1) 

x  '  y     a  v  ' 

In  like  manner,  we  shall  have 
1  (  11 
X     z      b 


-K=s (2) 


111 

-+-=- 3) 

y  '  z      c  v  ' 

Subtract  equation  (3)  from  (1), 

1111 

—  =—       (4) 

x     z      a     c  v  ' 

Add  equations  (2)  and  (4), 

2_1      1      1 
x     a     b     c  ' 
2a> 


tzc-j-  be  —  ab' 


170  ALGEBRA. 

In  like  manner, 

2a  be 
J     ab-\-bc — ac 

2abc 
■      ab-\-ac — be' 

Let  t  be  tho  time  in  which  they  could  finish  it  if  all  worked  together;  then, 
by  Prob.  8, 


/l      1      1\ 


2abc 
ab-^-ac-\-bc 

(16)  What  two  numbers  are  those  whose  difference  is  7  and  sum  33  ? 

Ans.  13  and  20 
(1?)  To  divide  the  number  75  into  two  such  parts  that  three  times  the 
greater  may  exceed  7  times  the  less  by  15. 

Ans.  54  and  21. 

(18)  In  a  mixture  of  wine  and  cidor,  |  of  the  whole  plus  25  gallons  was 
wine,  and  A  part  minus  5  gallons  was  cider;  how  many  gallons  were  there  of 
each  ? 

Ans.  85  of  wine,  and  35  of  cider. 

(19)  A  bill  of  $34  was  paid  in  half  dollars  and  dimes,  and  the  number  of 
pieces  of  both  sorts  that  were  used  was  just  100 ;  how  many  were  there  of 
each  1 

Ans.  60  half  dollars  and  40  dimes. 

(20)  Two  travelers  set  out  at  tho  same  time  from  New  York  and  Albany, 
whose  distance  is  150  miles ;  one  of  them  goes  8  miles  a  day,  and  the  other  7 ; 
in  what  timo  will  they  meet  ? 

Ans.  In  10  days. 

(21)  At  a  certain  election  375  persons  voted,  and  the  candidate  chosen  had 
a  majority  of  91 ;  how  many  voted  for  ei 

Ans.  233  for  one,  and  142  for  the  other. 

(22)  What  number  is  that  from  which,  if  5  be  subtracted,  -  of  the  remain- 
der will  be  40  ? 

Ans.  65. 

(23)  A  post  is  I  in  the  mud,  A  in  tho  water,  and  10  feet  above  the  water-. 
what  is  its  whole  length  ? 

Ai.s.  24  IV 

There  is  a  fish  whoso  tail  weighs  9  pounds,  his  head  weighs  as  much 
as  his  tail  and  half  his  body,  and  his  body  weighs  as  much  as  his  head  and  his 
Gail;  what  is  the  whole  weight  of  the  fish  .' 

(25)   After  |  way  '.  and  '  of  my  money.  !  had  66  guineas  left  in  my 

pane;  what  was  in  it  at  first? 

Ans.  120  guineas. 


SIMPLE  EQUATIONS.  171 

(26)  A's  age  is  double  of  B's,  and  B's  is  triple  of  C's,  and  the  sum  of  all 
their  ages  is  140;  what  is  the  age  of  each  ? 

Ans.  A's  =84,  B's  =42,  and  C's  =14. 

(27)  Two  persons,  A  and  B,  lay  out  equal  sums  of  money  in  trade ;  A 
gains  $630,  and  B  loses  $435,  and  A's  money  is  now  double  of  B's ;  what  did 
each  lay  out  ? 

•  Ans.  $1500. 

(28)  A  person  bought  a  chaise,  horse,  and  harness,  for  8450;  the  horse 
came  to  twice  the  price  of  tho  harness,  and  the  chaise  to  twice  the  price  of 
the  horse  and  harness ;  what  did  he  givo  for  each  ? 

Ans.  $100  for  the  horse,  $50  for  tho  harness,  and  $300  for  the  chaise. 

(29)  Two  persons,  A  and  B,  have  both  tho  samo  income :  A  saves  J  of  his 
yearly,  but  B,  by  spending  $250  per  annum  more  than  A,  at  the  end  of  4 
years  finds  himself  $500  in  debt ;  what  is  their  income? 

Ans.  $625. 

(30)  A  person  has  two  horses,  and  a  saddle  worth  $250 ;  now,  if  the  sad- 
dle bo  put  on  the  back  of  the  first  horse,  it  will  make  his  value  double  that  of 
the  second  ;  but  if  it  be  put  on  the  back  of  the  second,  it  Avill  make  his  value 
triple  that  of  the  first ;  what  is  the  value  of  each  horse  ? 

Ans.  One  $150,  and  the  other  $200. 

(31)  To  divide  the  number  36  into  three  such  parts  that  4  of  the  first,  i  of 
tho  second,  and  i  of  the  third  may  be  all  equal  to  each  other  ? 

Ans.  The  parts  are  8,  12,  and  16. 

(32)  A  footman  agreed  to  serve  his  master  for  c£8  a  year  and  a  livery,  but 
was  turned  away  at  the  end  of  7  months,  and  received  only  c-£2  13s.  4.d.  and 
his  livery  ;  what  was  its  value  1 

Ans.  c£4  16s. 

(33)  A  person  wras  desirous  of  giving  3d.  a  piece  to  some  beggars,  but  found 
that  ho  had  not  money  enough  in  his  pocket  by  8 d. ;  he  therefore  gave  them 
each  2d.,  and  had  then  3d.  remaining ;  required  tho  number  of  beggars  ? 

Ans.  11. 

(34)  A  person  in  play  lost  |  of  his  money,  and  then  won  3s. ;  after  which, 
he  lost  i  of  what  he  then  had,  and  then  won  2s. ;  lastly,  he  lost  j  of  what  he 
then  had;  and  this  done,  found  he  had  but  12s.  remaining;  what  had  he  at 
first? 

Ans.  20s. 

(35)  To  divide  the  number  90  into  4  such  parts  that  if  the  first  be  increased 
by  2,  the  second  diminished  by  2,  the  thud  multiplied  by  2,  and  the  fourth 
divided  by  2,  the  sum,  difference,  product,  and  quotient  shall  be  all  equal  to 
each  other  ? 

Ans.  The  parts  are  18,  22,  10,  and  40  respectively. 

(36)  The  hour  and  minute  hand  of  a  cle  xactly  together  at  12  o'clock  : 
when  are  they  next  together  ? 

Ans.  1  hour  5/,-  minutes. 

(37)  There  is  an  island  73  miles  in  circumference,  and  three  footmen  all 
start  together  to  travel  the  same  way  about  it :  A  goes  5  miles  a  day,  B  8,  and 
C  10;  when  will  they  all  come  together  again  ? 

Ans.  73  days. 


172  ALGEBRA. 

(38)  How  much  foreign  brandy  at  85.  per  gallon,  and  domestic  spirits  at  35. 
per  gallon,  must  be  mixed  together,  so  that,  in  selling  the  compound  at  9s.  per 
gallon,  the  distiller  may  clear  30  per  cent.  ? 

Ans.  51  gallons  of  brandy,  and  14  of  spirits. 

(39)  A  man  and  his  wife  usually  drauk  out  a  cask  of  beer  in  12  days  ;  but 
when  the  man  was  from  home,  it  lasted  the  woman  30  days  ;  how  many  daj  b 
would  the  man  alone  be  in  drinking  it? 

Ans.  20  days. 

(40)  If  A  and  B  together  can  perform  a  piece  of  work  in  8  days ;  A  and  C 
together  in  9  days  ;  and  B  and  C  in  10  days :  how  many  days  will  it  take 
each  person  to  perform  the  same  work  alone  ? 

Ans.  A  14fj  clays,  B  17;:;.  and  C  21 

(41)  A  book  is  printed  in  such  a  manner  that  each  page  contains  a  certain 
number  of  lines,  and  each  line  a  certain  number  of  letters.  If  each  pai:e  were 
required  to  contain  3  lines  more,  and  each  line  4  letters  more,  the  numb 
letters  in  a  page  would  be  greater  by  22 1  than  before ;  but  if  each  page  were 
required  to  contain  2  lines  less,  and  each  line  3  letters  less,  the  number  of  let- 
ters in  a  page  would  bo  less  by  145  than  before.  Required  the  number  of 
lines  in  each  page,  and  tho  number  of  letters  in  each  line. 

Ans.  29  lines,  32  letters. 

(42)  Hiero,  king  of  Syracuse,  had  given  a  goldsmith  10  pounds  of  gold  with 
which  to  make  a  crown.     The  work  being  done,  the  crown  was 

weigh  10  pounds;  but  the  king,  suspecting  that  the  workman  had  alio] 
with  silver,  consulted  Archimedes.     The  latter,  knowing  that 
water  52  thousandths  of  its  weight,  and  silver  99  thousandths,  ascertained  the 
weight  of  tho  crown,  plunged  in  water,  to  be  9  pounds  6  ounces.     This  dis- 
covered tho  fraud.     Required  the  quantity  of  each  metal  in  the  crown. 

Ans.  7  pounds  12}§  ounces  of  gold,  2  pounds  3^  ounces  of  silver. 

(43)  To  divide  a  number  a  into  two  parts  which  shall  have  to  each  other 
the  ratio  of  m  to  n. 

ma        na 

Ans. : — ,  — ; — . 

m-\-n  m-\-n 

(44)  To  divide  a  number  a  into  three  parts  which  shall  be  to   each  ether 

as  m:n:p. 

ma  7i  a  pa 

Ans. 


7ll-\-n-\-J>,  in  -\-n  -\-j>'  m  -\-  n  -\-p 

(45)  A  banker  has  two  kinds  of  change  ;  there  must  be  a  pieces  of  the  first 

to  make  a  crown,  and  h  pieces  of  the  second  to  make  the  same  :  now  a  per- 
son wishes  to  have  c  pieces  for  a  crown.     How  many  pieces  of  each  kind  must 

the  banker  give  him? 

a{b — c)  /'(c — a)    „  .  , 

Ans.  -r -  of  the  firsl  kind.  — ; of  the  second. 

h —  a  b — a 

(46)  An  innkeeper  makes  this  bargain  with  a  Bportsman:  every  day  that 
the  latter  brings  a  certain  quantity  of  game  he  is  to  receive  a  sum  a,  but  every 
day  that  lie  fails  to  bring  it  he  is  to  pay  a  sum  6.  After  a  number  n  of 
days  it  may  happen  that  neither  owes  the  other,  or  that  the  first  ewes  the 
stM 1,  01  that  ti  id  owes  the  firsl  a  Bum  c.     Required  if  formula  which 


SIMPLE  EQUATIONS.  17.? 

snail  express  in  all  three  cases  the  number  of  days  that  the  sportsman  brought 
the  game. 

Ans.  x= — — 7-. 
a-\-u 

In  the  first  case  c=0,  in  the  second  case  we  must  take  the  positive  sign,  in 

the  third  case  tho  negative  sign. 

(47)  If  one  of  two  numbers  be  multiplied  by  m,  and  the  other  by  n,  the  sum 
of  the  products  is  p !  but  if the  m*st  be  multiplied  by  m',  and  the  second  by  n', 
the  sum  of  the  products  is  p'.     Required  the  two  numbers. 

n'p — np'    mp' — m'p 

Ans.  : t;  • ; 7~« 

mm! — m'n   mn' — m'n 

(48)  An  ingot  of  metal  which  weighs  n  pounds  loses  p  pounds  when  weigh 
ed  in  water.     This  ingot  is  itself  composed  of  two  other  metals,  which  we 
may  call  M  and  M' ;  now  n  pounds  of  M  loses  q  pounds  when  weighed  in 
water,  and  n  pounds  of  M'  loses  r  pounds  when  weighed  in  water.     How 
much  of  each  metal  does  the  original  ingot  contain  ? 

Ans.  !^J-P)  pounds  0f  M,  n{p~q)  pounds  of  M'. 

r — q  r — q      * 

REMARKS  UPON  EQUATIONS  OF  THE  FIRST  DEGREE. 
152.  Algebraic  formulae  can  offer  no  distinct  ideas  to  the  mind  unless  they 
represent  a  succession  of  numerical  operations  which  can  be  actually  perform- 
ed. Thus,  the  quantity  b  — a,  when  considered  by  itself  alone,  can  only  sig- 
nify an  absurdity  when  a>6.  It  will  be  proper  for  us,  therefore,  to  review 
the  preceding  calculations,  since  they  sometimes  present  this  difficulty. 

Every  equation  of  the  first  degree  may  be  reduced  to  ono  which  has  all  it* 
signs  positive,  such  as 

az-\-b=cx-\-d (1)* 

Subtracting  c.r+6  from  each  member,  we  then  have 

ax — ex=d — b. 
Whence 

a:= (2) 

a—  c  ' 

This  being  premised,  three  different  cases  present  themselves  ; 

1°.  c£>6  and  a>c. 

2°.  One  of  these  conditions  only  may  hold  good. 

3°.  b>d  and  c>a. 

In  the  first  case  the  value  of  x  in  equation  (2)  resolves  the  problem  without 
giving  rise  to  any  embarrassment ;  in  the  second  and  third  cases  it  does  not,  at 
first,  appear  what  signification  we  ought  to  attach  to  the  value  of  x ;  and  it  is 
this  that  we  propose  to  examine. 

In  the  second  case  one  of  the  subtractions,  d — b,  a — c,  is  impossible ;  for 
example,  let  b>d  and  a~>c;  it  is  manifest  that  the  proposed  equation  (1)  is 
absurd,  since  the  two  terms  ax  and  b  of  the  first  member  are  respectively 
greater  than  the  two  terms  ex  and  d  of  the  second.  Hence,  when  we  en- 
counter a  difficulty  of  this  nature,  we  may  be  assured  that  the  proposed  prob- 

*  We  can  always  change  the  negative  terms  of  an  equation  into  positive  ones  by  trans- 
posing them  from  the  member  in  which  they  arc  found  to  the  other  member. 


171  ALGEBRA. 

lorn  is  absurd,  since  the  equation  is  merely  a  faithful  expression  of  its  condi- 
tions in  algebraic  language. 

In  the  third  case  we  suppose  b^>d  and  r^>a;  here  both  subtractions  are 
impossible  ;  but  let  us  observe  that,  in  order  to  solve  equation  (1),  we  subtract- 
ed from  each  member  the  quantity  cx-\-b,  an  operation  manifestly  impossible, 
since  each  member  <^cx-\-b.     This  calculation  being  erroneous,  let  us  sub 
tract  ax-\-d  from  each  member  ;  we  then  have 

b — d=cx — ax. 

Whence 

b-d 
x= (3) 

c — a  v  ' 

This  value  of  x,  when  compared  with  equation  (2),  differs  from  it  in  this 
only,  that  the  signs  of  both  terms  of  the  fraction  have  been  changed,  and  the 
solution  is  no  longer  obscure.  "We  perceive  that,  when  we  meet  with  this 
third  case,  it  points  out  to  us  that,  instead  of  transposing  all  the  terms  involv- 
ing the  unknown  quantity  to  tho  first  member  of  the  equation,  we  ought  to 
place  them  in  the  second  ;  and  that  it  is  unnecessary,  iu  order  to  correct  this 
error,  to  recommence  the  calculation ;  it  is  sufficient  to  change  the  signs  of 
both  numerator  and  denominator. 

When  the  equation  is  absurd,  as  in  the  second  case,  we  may  nevertheless 

make  use  of  tho  negative  solution  obtained  in  this  case  ;  for  if  we  substitute 

— x  for  -\-x,  tho  proposed  equation  becomes 

— ax-\-  b  s=  — cx-\-  d. 

b—d 

Whence  ,r= , 

a — c 

a  value  equal  to  that  in  (2),  but  positive.     If,  then,  we  modify  the  question  in 

such  a  manner  as  to  agree  with  this  new  equation,  this  second  problem,  whicl 

will  bear  a  marked  resemblance  to  tho  first,  will  no  longer  be  absurd,  and. 

with  the  exception  of  the  sign,  will  have  the  same  solution. 

Let  us  take,  for  example,  the  following  problem  : 

A  father,  aged  42  years,  has  a  son  aged  12  ;  in  how  many  years  will  the  age 
of  the  son  be  one  fourth  of  that  of  the  father? 

»    Let  .r=  the  number  of  years  required. 

42+3 
Then  — j- =12+*; 

...  x— o. 

Thus  the  problem  is  absurd.  But  if  wo  substitute  — x  for  -\-r,  the  equa- 
tion becomes 

— x 
=12 r 

4 
and  tho  conditions  corresponding  to  this  equation  change  the  problem  to  the 
following : 

A  father,  aged  42  years,  has  a  son  aged  12  ;  how  many  years  have  elapsed 
sinrc  the  ago  of  the  son  was  one  fourth  of  that  of  the  father  !* 
Here  -r— 

"  As  a  problem  is  translated  into  algebraic  !  by  means  of  an  equation,  so  as 

equation  maybe  translated  back  into  a  problem,  provided  the  general  uaturoof  the  probl*'  ■ 
be  kivowu. 


SIMPLE  EQUATIONS.  175 

Take  another  example. 

What  number  of  dollars  is  that,  the  sum  of  the  third  and  fifth  parts  of  which, 
diminished  by  7,  is  equal  to  the  original  number  ? 

x     x 
Here  -+--7=.r. 

Whence  x=— 15. 

The  problem  is  absurd  •  but,  substituting  — x  for  +.r, 

x     x 
-3-5-7  =  -*; 
or 

x     x 

3+5X7=*' 

which  gives 

a:=15; 

and  the  problem  should  read,  What  number  of  dollars  is  that,  the  third  and  fif  tl»  ' 
parts  of  which,  when  increased  by  7,  give  the  original  number  ? 

153.  With  regard  to  the  interpretation  of  negative  results  in  the  solution 
of  problems,  then,  wo  may,  from  what  is  seen  above,  establish  the  following 
general  principle  : 

When  we  find  a  negative  value  for  the  unknown  quantity  in  problems  of  the 
first  degree,  it  points  out  an  absurdity  in  the  conditions  of  the  problem  pro- 
posed ;  provided  the  equation  be  a  faithful  representation  of  the  problem,  and 
of  the  true  meaning  of  all  the  conditions. 

The  value  so  obtained,  neglecting  its  sign,  may  be  considered  as  the  answer 
to  a  problem  which  differs  from  the  one  proposed  in  this  only,  that  certain  quan- 
tities which  were  additive  in  the  first  have  become  subtractive  in  the  second,  and 
reciprocally. 

154.  The  equation  (2)  presents  still  two  varieties.     If  a=c,  we  have 

d-b 
*=— ; 

in  this  case  the  original  equation  becomes 

ax-\-b  =  ax-{-d, 
whence  b=d  ;  if,  therefore,  b  be  not  equal  to  d,  the  problem  would  seem  ab- 
surd.* 

d—b         .  ,  m 

But  the  expression  — - — ,  or,  m  general,  — ,  where  m  may  be  any  quantity, 

m 
represents  a  number  infinitely  great.     For,  if  we  take  a  fraction  — ,  the  small- 


er we  make  n,  the  greater  will  the  number  represented  by  —  beccme;  thus, 


n 

m 
n 


for  n=->  -rr-z,  Tz-rzi  the  results  are  2,  100,  1000  times  m.     The  limit  is  in- 


in- 


finity, wliich  corresponds  to  n=0.     Or,  we  may  say,  to  prove  —  infinite,  that 

*  The  absurdity  is  removed  by  considering  that  finite  quantities  have  no  effect  when 
added  to  infinite  ones ;  that,  in  comparison  with  infinities,  finite  quantities  are  all  equal  to 
one  another,  and  all  equal  to  zero 


176  ALGKBKA. 

a  finite  quantity  evidently  contains  an  infinite  number  of  zeros.     The  symbol 
for  the  value  of  x  in  this  case  is 

By  clearing  the  expression  77=00  of  fractions,  wo  have  ??i=0Xoo>  from 

TO 

which  it  appears  that  the  product  of  zero  by  infinity  is  finite.     So,  also,  — =0, 

or  the  quotient  of  a  finite  quantity  by  infinity,  is  zero. 

155.  If,  in  equation  (2),  a=c,  and  b=d,  we  have 

0 

in  this  case  the  original  equation  becomes 

ax-\-b=.ax-^-b. 
Here  the  two  members  of  the  equation  are  equal,  whatever  may  be  the  value 
of./-,  which  is  altogether  arbitrary,  and  may  have  any  value  at  pleasure.     We 
perceive,  then,  that  a  problem  is  indeterminate,  and  is  susceptible  of  an  in- 
finite number  of  sohdions,  when  the  value  of  the  unknoicn  quantity  appears 

under  the  form  -. 

0 
It  is,  however,  highly  important  to  observe,  that  the  expression  -  does  not 

always  indicate  that  the  problem  is  indeterminate,  but  merely  the  existence  of 

a  factor  common  to  both  terms  of  the  fraction,  which  factor  becomes  0  under 

a  particular  hypothesis. 

Suppose,  for  example,  that  the  solution  of  a  problem  is  exhibited  under  the 

a5— b3 
forra  x=-^-^. 

If,  in  this  formula,  we  make  a  =  &,  then  r=-. 

771 

*  This  infinite  value  of  expressions  liko  —  may  be  sometimes  positive,  sometimes  nega 
tive,  and  sometimes  indifferently  positive  or  negative. 
1°.  Let  there  be  the  formula  x-=- -,  in  which  7n  and  11  are  two  invariable  numbers. 

which  we  suppose  positive,  and  different  from  zero,  while  -  can  have  all  possible  values. 

in 
Making  z=n,  we  have  .t=--.    But  as  the  denominator,  («—  .ays  positive,  what- 

ever1 z  may  bo,  the  infinity  here  should  be  regarded  as  designating  the  positive  infinity. 

—771 

S3.  By  analogous  reasoning,  we  see  that  if  we  have  the  formula  x=—  and  *=« 

should  have  the  negative  r= — <». 

3J.  Let  there  be  the  formula  x= .     Tl  till  j=— ,  but  hen 

n — z  '  0 

the  infinity  will  have  an  ambiguous  sign.     Sup;  and  cause  -  to  incr 

the  formula  will  give  increasing  values,  which  will  be  all  1  On  the  contrary,  takin 

z>n,  then  diminishing  Z  till  it  becomes  equal  to  n,  the  formula  gives  increasing  \; 

'.'.  Inch  are  negative.    Therefore,  the  hypothesis  z=n  ought  to  be  considered  as  causir 

formula  to  take  two  infinite  values,  the  one  positive  and  the  other  negative.     This  i 

cat  :d  by  writing  x=~L<».    Tho  00  is  here  tho  transition  value  between  -f-  and — .    Zen 

1  a  1 ransition  value  between  -\-  and  — .    For,  let  x=n — X  :  if  r<»,  and  s  incre: 

2>n,  tho  value  of  x  in  changing  from  -f-  to  —  1  rough  0.     Quantities  in  eh. 

#ign  must  always  pass  through  0  or  00  .     '1  .  pass  through  0  or  00  with 

out  changing  si-n,  as  in  *=(n — s)",  and  ; r. 


PIMPLE  EQUATIONS.  17? 

But  we  must  re  nark,  that  a? — b3  may  be  put  under  the  form  (a  —  b) 

(a?-{-ab-{-b"),  and  that  a"—b2  is  equivalent  to  (a  —  b)  (a-\-b) ;  hence   the 

above  value  of  x  will  be 

(fl  — 6)(a«+a&+68) 

X~      (£"— 6)(a  +  6)      ' 
Now  if,  before  making  the  hypothesis  a  =  6,  wo  suppress  the  common  fac- 
tor a  —  b,  the  value  of  x  becomes 

a*+ab  +  b* 
x=      a~+b      ' 
an  expression  wh/ch,  under  the  hypothesis  that  «  =  &,  is  reduced  to 

3a2     3a 
a?s=2aa="2* 
Take,  as  a  second  example,  the  expression 

at  —  b*      (a  +  b){a—b) 


x= 


(a  —  6)3— (a  — b){a— 6)' 
0 


making  #  =  &,  the  value  of  x  becomes  £==-,  in  consequence  of  the  existence 

of  the  common  factor  a—  b  ;  but  if,  in  the  first  instance,  we  suppress  the  com- 
mon factor  a — J,  the  valuo  of  .r  becomes 

a+b 
X—~^—b; 
sn  expression  which,  under  Vhe  hypothesis  that  a=b,  is  reduced  to 

2a 
z=-=co. 

From  this  it  appears  that  the  symbol  -  in  algebra  sometimes  indicates  die 

existence  of  a  factor  common  to  the  two  terms  of  the  fraction  which  is  reduced  to 
that  form.  Hence,  before  we  can  pronounce  with  certainty  upon  the  true 
value  of  such  a  fraction,  we  must  ascertain  whether  its  terms  involve  a  com- 
mon factor.  If  none  such  be  found  to  exist,  then  we  conclude  that  the  equa- 
tion in  question  is  really  indeterminate.  If  a  common  factor  be  found  to  exist, 
we  must  suppress  it,  and  then  make  anew  the  particular  hypothesis.  This 
will  now  give  us  the  true  value  of  the  fraction,  which  may  present  itself  under 

A  A   0 
one  of  the  three  forms  ^,  — ,  -. 

In  the  first  case,  the  equation  is  determinate ;  in  the  second,  it  is  impossible 

in  finite  numbers  ;  in  the  third,  it  is  indeterminate. 

0 
There  are  other  forms  of  indetermination  besides  -  ;  for,  whatever  be  the 

values  of  P  and  Q,  wo  have 

1_ 

P  2._§ 

Q-   xQ~r 

P 
P 

The  first  of  these  equivalents  of  j=r,  where  P  and  Q  both  equal  zero,  be- 

00 

comes  OX00)  and  the  second  becomes  — ,  which  symbols  must,  therefore,  be 

OD 

0 
considered  as  having  the  same  meaning  with  -. 

M 


i;-  ALGEBJ 

i.N    OF   FORMULAS   FURNISHED    BT   Ti.  BRAL  EQUATIONS    OF  TB* 

FIRST    DECREE,    WITB    TWO    OR    MORE    I  I  N    QJ0ANTIT1 

When  the  common  denominator  of  the  general  values  of  die  unknown  quan 
tities  reduces  to  zero,  it  is  not  readily  seen  how  the  given  equations  are  to  be 
verified.     We  shall  examine  here  the  particular  cases  of  this  kind  which  may 
occur. 

Resume  the  two  equations, 

a  x+b  y=k  [1] 

:+b'yz=k'  [2] 

from  which  we  derive  the  formulas 

Jcb'  —  bk'         ale'— Tea' 
X=  ab'  —  ba"  V=  ab'  —  ba'' 
First  particular  Case. — Suppose  the  denominators  to  be  z  the  uu 

merators  not :  then  we  have 

:  —  bk'           aJc'—l 
ab'  —  ba1 =  0,  x=z ,  y= . 

The  values  of  x  and  y  are  then  infinite  ;  that  is  to  say,  in  order  to  satisfy  the 
two  given  equations,  they  must  surpass  every  assignable  magnitude. 

From  the  equality  ab'  —  ba'=0,  we  derive  a'=-j-,  and,  consequently,  the 

equation  [2],  by  putting  in  it  this  value,  becomes 

ab' 

-j-X-\-b'y  =  lc\  .-.  b'(ax-\-by)  =  l 

,     The  first  member  is  the  first  member  of  [1]  multiplied  by  1/  ;  the  same  re- 
lation must  subsist  between  the  second  members,  in  order  that  the  value  of  a 
and  y  may  verify  at  the  same  time  equations  [1]  and  [2].     Heuce  bk'=kb', 
or,  kb' — bk'=0  ;  i.  c,  the  numerator  of  x  would  be  equal  to  zero,  wh'u 
contrary  to  hypothesis.* 

In  this  way  the  impossibility  of  finding  values  of  x  and  y,  which  satisfy  at 
the  same  time  the  two  given  equations,  is  made  apparent;  but  this  im; 
bilily  is  siill  better  characterized  by  the  infinil  .  which,  at  the  same  time 

that  they  indicate  the  impossibility,  show  besides  that  it  arises  from  the  fact 
thai  the  values  of  the  unknown  quantities  are  t->o  great  to  bo  as 

[f  we  suppose  ab'  —  ba'  to  bo  at  first  a  tall  quantity,  tho  values  of  x 

and  y  will  be  very  great,  but  they  will  alway  '    the  equati  ins  until  the 

instant  ab'  —  ba'  reduces  to  zero,  when,  if  ct  in  a  direct  mannei 

the  verification  of  the  equations,  it  is 
ignable  magnitude.! 

Second  particular  Case. — Suppose  the  denominator  to  he  zero  at  the  same 
time  as  one  of  the  numerators;  for  example,  thai  w< 

ab'—ba'=0t  W  —  /■/.■■  =  0. 

I  maintain  that  the  other  numerator  will  be  also  equal  to  zero;  for  th» 
two  equ  Jities  aboi  e  give 

*  The  note  to  art          i                  anomaly.    The  tantitiea  kV  and  W  arc  i 
shen  compared  with  ii  finity. 

t  Considered  in  relation  i<<  the  question,  lie  conditions  of  which  nr.  I  j  tho 

problem,  infinil                may  1><-  sometimes  :i  Irae  solot  ition.    She  applies 

ii,„,  of                              try  furnishes  nam  th  ra  may 

b,  cited  th  'i  \\  bare  a  i  angle  ii  onknown,  and  we  find  for  its  tangent  an  infinite  vata 
in  clear,  tli.n.  thai  thl  an  |le  mnat  be  right 


SIMPLE  EQUATIONS.  179 

al'      _kV 

a'— T'  b  ' 

and,  consequently,  the  other  numerator  becomes 

akb'     alcb' 
ak' — ka'= — ; — — — ; — =0 
b  b 

Jf  at  first  we  had  supposed  this  numerator  equal  to  zero.,  we  could  have 

proved  in  a  similar  manner  that  of  a:  to  be  so  also. 

The  present  hypothesis  then  gives 

0  0 

Of  themselves  these  symbols  indicate  indetermination ;  I  shall  prove,  by  going 
back  to  the  equations,  that  they  ought,  in  fact,  to  be  indeterminate. 

For  this  purpose,  substitute  in  equation  [2]  the  values  of  a'  and  k',  found 
■ibove,  and  it  becomes 

aV       ,        W       V.      ,  _   x     b' 

-yZ+i'7/  =  — ,  .:^{ax+b7j)=jk. 

Thus  we  see  that  it  can  be  formed  by  multiplying  the  two  members  of  equa- 

b' 
tion  [1]  by  -r- ;  then  all  values  of  x  and  y  which  satisfy  one  of  the  two  equations 

will  also  satisfy  the  other.    But  if  we  give  to  x  values  at  pleasure  in  equation  [1], 
an,  by  resolving  it  afterward,  find  corresponding  values  of  y  ;  aud  as  these 
same  values  satisfy  the  second  equation,  we  conclude  that  the  proposed  equa 
lions  admit  an  infinite  number  of  solutions. 

Let  it,  however,  be  observed,  that  the  indetermination  in  this  case  does  not 
permit  us  to  take  whatever  value  of  y,  and,  at  the  same  time,  of  x,  we  please, 
because  tho  above  explication  shows  that,  when  one  of  these  unknown  quan- 
tities is  assumed,  the  value  of  the  other  is  determined. 

The  case  before  us  comprehends  that  in  which  k=0,  k' =0,  ab'—ba'=0, 

because  then  x  and  y  become  -.     If  we  return  to  the  equations  proposed,  they 

reduce  to  these, 

ax-\-by=0,  afx-{-b'y=0. 

They  give  respectively 

a  a' 

y=—bx,y  =  -vx. 

a     a' 
But  upon  the  hypothesis  of  ab' —  ba'=0,  we  derive  t=j-,;  then  the  two 

values  of  y  are  equal,  whatever  be  that  of  x,  and  there  is  veritable  indeter- 
mination. 

Yet  it  is  to  be  observed,  that,  if  we  take  the  relation  of  y  to  x,  this  relation 
is  determinate,  because  we  have 

y         a         a' 

x=~b=~b'' 

If  tho  condition  7=77  had  not  existed,  the  two  values  of  y  above  could  wot 

have  been  equal,  except  we  suppose  .r=0  ;  y  would  have  been  then  zero,  and 
the  relation  of  x  and  y  no  longer  determinate,  but  indeterminate. 

A  similar  discussion  to  the  above  might  be  given  to  a  system  of  three  or  more 
equations,  with  as  many  unknown  quantities.  It  would,  however,  be  more 
rlilficult  to  investigate  the  cases  of  impossibility  and  iudetermination,  and  it  is 


180  ALGEBRA 

not  worth  while  to  delay  upon  them.  We  shall  content  ourselves  with  setting 
down  here  some  observations  intended  to  caution  the  student  against  certaiD 
hasty  conclusions  to  which  he  might  naturally  be  led. 

We  have  seen,  in  the  case  of  two  equations  with  two  unknown  quantities 
that  x  and  y  become  infinite  and  indeterminate  simultaneously. 

The  first  error  which  might  be  committed  would  be  that  of  supposing  from 
analogy  that,  in  the  case  of  several  equations,  the  unknown  quantities  would 
all  become  infinite  or  indeterminate  together.  Suppose,  for  example,  then* 
are  under  consideration  the  three  equations 

ax  -\-by  -\-cz   =/c, 
a'x  -j-o'y  +c'z  =ld, 
a"x+b"y+c"z=k". 
The  common  denominator  of  the  values  of  r,  y,  z,  is 

R=ab'c"  —  ac'b"-\-ca'b"  —  ba'c"+bc'a"— cb'a' 
and  it  may  be  written  in  three  ways : 

R=a{b'c"—c'b")  +a'{cb"  —bc")  +  a"{bc'—cb'y 
R=b{c'a"—a'c")+b'(ac"  —  ca")  +  b"(ca'—ac'y 
R=c(a'b"  —  b'a")  +  c'(ba"—ab")  +  c"(ab'—li') 
Place 

b'c"=c'b",  cb"  =  bc". 

From  these  equations  we  deduce  bc'=cb',  and,  consequently,  R  becomes 

eero.     Then  the  numerator  of  .r,  which  is  formed  from  R  by  changing  a,  a'. 

•a"  into  k,  k',  k",  becomes  zero  also.     But  as  the  numerator  of  y  is  formed  by 

placing  k,  k',  k"  in  R  instead  of  b,  6',  b",  there  is  no  reason  why  this  numerator 

should  become  zero,  unless  we  make  some  new  hypothesis.     The  same  may 

be  said  of  that  of  z.     Thus  the  value  of  .r  can  take  the  indeterminate  form  -, 

0' 

where  the  values  of  y  and  z  are  infinite. 

But  with  regard   to  this  indeterminate  form,  another  error  still  is  to  be 

avoided,  because  it  may  bo  that  the  indetermination  is  only  apparent  (see 

Art.  153).     In  order  to  judge  better  of  it,  we  shall  have  regard  only  to  th*» 

single  relation 

c'b" 
b'c"=c'b",  .:c"=-rr. 

Substituting  this  valuo  of  c"  in  the  general  value  of  r,  it  will  be  seen  that 
be' — cb'  becomes  a  common  factor  of  both  numerator  and  denominator.  But 
by  hypothesis  this  factor  is  zero:  it  i    i  ace,  t!  en.  which  produces  the 

appearance  of  indetermination.     Suppressing  it,  we  havt  the  true  value  of  .r, 
which  appears  no  longer  indeterminate  some  new  hypothesis  be  joined 

to  those  already  made.* 

*  An  important  observation  should  bo  made   bel  the  subject  of  indetermi. 

nation. 

When  the  two  terms  of  n  fraction  decrease  so  as  to  1  than  any  assignable 

quantity,  if  the  suppositions  which  cause  one  of  I  entirely 

independent  of  those  which  cause  the  other  (•>  i  of  these  t 

at  zero  ii 

fraction,  maybe  c<]ual  to  ajiy  quantity  w 

arrhe  when  the  two  terms  shall  1  I  "f  ih.ir  •  '  will  expros» 

aplete  Indetermination.    Bnl  it  may  happen  that  tl  •   rms  of  the  fraction  are 

naeted  together  ii  inch  a  way.  that  to  a  < 


SIMPLE  EQUATIONS.  181 

i56.  We  shall  conclude  this  discussion  with  the  following  problem,  which 
will  serve  as  an  illustration  of  the  various  singularities  which  may  present 
themselves  in  the  solution  of  a  simple  equation. 

problem. 
Two  couriers  set  off  at  the  same  time 

from  two  points,  A  and  B,  in  the  same     — -pr, \ 4 7^ — 

•  i      ,•  j  ii  i-  ^  A         t$  U 

straight  line,  and  travel  in  the  same  di- 
rection, A  C     The  courier  who  sets  out  from  A  travels  m  miles  an  hour,  the 
courier  who  sets  out  from  B  travels  n  miles  an  hour ;  the  distance  from  A  to 
B  is  a  miles.      A.t  what  distance  from  the  points  A  and  B  will  the  couriers  be 
together? 

Let  C  be  the  point  where  they  are  together,  and  let  x  and  y  denote  the  dis- 
tances A  C  and  B  C,  expressed  in  miles. 

We  have  manifestly  for  the  first  equation 

x—y=a         .  (1) 

Since  m  and  n  denote  the  number  of  miles  traveled  by  each  in  an  hour,  that 

s.  the  respective  velocities  of  the  two  couriers,  it  follows  that  the  time  re- 

x    y 
mired  to  traverse  the  two  spaces,  x  and  y,  must  be  designated  by  — ,  -  ;  these 

J  &  J  m   n 

two  periods,  moreover,  are  equal ;  hence  we  have  for  our  second  equation 
.t      y 
m~n '  *  ' 

The  values  of  x  and  y,  derived  from  equations  (1)  and  (2),  are 

am  an 


x= ,  y= . 

1°.  So  long  as  we  suppose  m^>n,  or  m — n  positive,  the  problem  will  be 
solved  without  embarrassment.  For,  in  that  case,  we  suppose  the  courier  who 
starts  from  A  to  travel  faster  than  the  courier  who  starts  from  B ;  he  must, 
therefore,  overtake  him  eventually,  and  a  point  C  can  always  be  found  where 
they  will  be  together. 

2°.  Let  us  now  suppose  m<«,  or  m  —  n  negative,  the  values  of  a:  and  y  are 
both  negative,  and  we  have 

am  an 


.r= ,  y=  — . 

The  solution,  therefore,  in  this  case,  points  out  that  some  absurdity  must  exist 
in  the  conditions  of  the  problem.  In  fact,  if  we  suppose  m<^n,  we  suppose 
tliat  tho  courier  who  sets  out  from  A  travels  slower  than  the  courier  who  seta 
out  from  B ;  hence  the  distance  between  them  augments  every  instant,  and  it 
is  impossible  that  the  couriers  can  ever  be  together  if  they  travel  in  the  di- 
rection A  C.  Let  us  now  substitute  — x  for  -f-.r,  and  — y  for  -{-y,  in  equa- 
tions (1)  and  (2) ;  when  modified  in  this  manner,  they  become 

a  very  .small  value  of  the  other ;  and  that,  when  they  converge  toward  zero,  their  relation 
converges  toward  a  determinate  limit,  which  it  does  not  attain  till  the  moment  that  the 

0 
two  terms  vanish,  and  the  fraction  presents  itself  under  the  form  -.*    A  particular  exam- 
ple of  this  last  case  is  the  vanishing  of  a  common  factor  of  the  numerator  and  denominator 
Th»  same  remark  is  applicable  to  the  svmbol  — . 

00 

*  Ti..  principle  is  fully  exemplified  in  the  differential  calculun 


i£2  ALGEBRA.. 

y—x  =  a  \ 

-  =  -( 
m      11  ) 

equations  which,  when  resolved,  give 

din  an 


r= ,  y= . 

in  which  llie  values  of  x  and  y  are.  positive. 

These  values  of  x  and  y  give  the  solution,  not  of  the  proposed  problem 
which  is  absurd  under  the  supposition  that  ?«<«,  but  of  the  following,  which 
is  llie  translation  of  the  changed  equati< 

Two  couriers  set  out  at  iho  same  time  from  tho  points  A  and  B,  and  travel 
in  t  B  C,  tVc.  (tho  rest  as  before)  ;  the  values  of  .r  and  y  mark  the 

distances  A  C,  B  C,  of  the  point  C,  where  the  couriers  are  together,  from 
the  points  of  departure  A  and  B. 

From  this  problem,  as  well  as  that  of  the  father  and  son  above,  may  be  de- 
duced the  following  rule,  when  the  value  of  the  unknown  quantity  is  found  to 
be  negative  : 

Change  (he  sign  of  the  unknown  quantity  in  the  first  equation,  or  the  one 
deri  'iately  from  the  probh  'nation,  translated  into 

common  lung"       .      Ill  furnish  the  problem  which  will  give  a  positive  solut 

If  the  j  he  at  first  enunciated  in  a  general  manner,  then  negative 

values  of  the  unknown  quantity  may  he  regarded  as  furnishing  a  true  s 
but  are  to  be  interpreted  in  a  contrary  sense.     Thus,  if  positive  values  repre- 
sent distance  to  the  right,  negative  will  rejjrcscnt  distance  to  the  I  si- 
tive  express  elistance  upward,  negative  distance  downward;  if  (It e  for 
die*             future,  the  latter  must  indicate  time ])ast ;  if  the  one  gain,  the  o' 
loss ;  if  the  one  a  rate  of  increase,  the  other  a  rate  of  decrease,  eye* 

3°.  Let  us  next  suppose  m=u  ;  the  values  of  x  and  y  in  this  case  becom« 

am         an 

or 

r=oo ,  2/  =  °° ; 

that  is  to  say,  x  ami  y  each  represent  infinity.  In  fact,  if  we  suppose  m  =  n, 
we  suppose  the  coi  i  sets  out  from  A  to  travel  exactly  at  the  same  rate 

as  the  courier  who  sets  out  from  I!  ;  consequently,  the  original  distance,  a,  by 
which  they  are  separated  will  always  remain  the  same,  and  if  tho  couriers 
travel  fori  vi  r,  they  can  never  be  together.! 

*  Applications  of  this  use  of  positive  and  e  quantities  constantly  occur  in  trigtt- 

uomi  i 

.  n-=J.     Ti>  im- 
ir  t"  the  principle  stated  in  We  a  id  a 
littii                                                All  zen                                          ed  with  finite  quantities, 
but  i                            d  with  oi                    I                                     t  as  x,  though  x  be  0 : 
but  Zx-\-a=x-\-a=a,  if£=0.     I  i  the  li                                                         ■    mpared  with 

1 :  in  the  .       and  .r,  a 

with  i  qual. 

•i,  x-\-a  t  as  "■.  when  oa 

il  with  infinil  other,  and  all 

ed  v,  iili  .  "t  when  simp 

nnothi  !•.     h 


SIMPLE  EGIUATICLVS.  183 

4°.  Let  us  suppose  m=n,  and  also  a  =  0  ;  the  values  of  x  and  y  iu  this  case 
hecome 

0  0 

that  is  to  say,  the  problem  i  -tin ate,  and  admits  of  an  infinite  number 

of  solutions.  In  fact,  if  wo  suppose  «  =  0,  we  suppose  that  the  couriers  start 
from  the  same  point,  and  if  we  at  the  same  time  suppose  m=n,  or  that  they 
travel  equally  fast,  it  is  manifest  that  they  must  always  be  together,  and  conse- 
quently every  point  in  the  lino  A  C  satisfies  the  conditions  of  the  problem. 

•")'.  Finally,  if  we  suppose  a  =  0,  and  m  not  =n,  the  values  of  X  and  y  in 
this  case  becomo 

x=0,  t/  =  0. 

In  fact,  if  wo  suppose  the  couriers  lo  set  out  .from  tlio  same  point,  and 
to  travel  with  different  velocities,  it  is  manifest  that  the  point  of  departure  is 
the  only  point  in  which  they  can  be  together. 

ADDITIONAL    PROBLEMS. 

(1)  The  rent  of  an  estate  is  greater  than  it  was  last  year  by  8  per  cent,  ot 
tlio  rent  of  that  year  ;  this  year's  rent  is  1890.     What  was  last  year's  ? 

Ans.  1750. 

(2)  A  company  of  90  persons  consists  of  men,  women,  and  children  ;  th* 
men  are  4  in  number  more  than  the  women,  and  the  children  10  more  than 
the  men  and  women  together.     How  many  of  each  ? 

Ans.  22  men,  18  women,  and  50  children. 

(3)  From  the  first  of  two  mortars  in  a  battery  36  shells  are  thrown  before 
the  second  is  ready  for  firing.  Shells  arc  then  thrown  from  both  in  the  pro- 
portion of  8  from  the  first  to  7  of  the  second,  the  second  mortar  requiring  a* 
much  powder  for  3  charges  as  the  first  does  for  4.  It  is  required  to  deter- 
mine after  how  many  discharges  of  the  second  mortar  the  quantity  of  powder 
consumed  by  it  is  equal  to  the  quantity  consumed  by  the  first. 

Ans.  189  discharges  of  the  second  mortar. 

(4)  The  fore  wheels  of  a  carriage  are  5}  feet  and  the  hind  wheels  7 -J  feet 
in  circumference  ;  the  difference  of  the  number  of  revolutions  of  the  wheels 
is  2000.     What  is  the  length  of  the  journey  ? 

Ans.  39900  feet,  or  7$  miles. 

(5)  Three  brothers,  A,  B,  and  C,  buy  a  house  for  t£2000  ;  C  can  pay  the 
whole  price  if  B  give  him  half  his  money  ;  B  can  pay  the  whole  price  if  A 
give  him  one  third  of  his  money;  A  can  pay  the  whole  price  if  C  give  him 
one  fourth  of  his  money.     How  much  has  each  ? 

Ans.  A  c£l680,  B  <£1440,  C  ,£1280. 

(6)  The  passengers   of  a  ship  were   ]•  Germans,  I  French,  £  English,  I 

but  the  objection  to  the  doctrine  of  tlio  special  and  immediate  superintendence  of  Provi- 
lenc  Fairs  of  men,  thnt  it  implies  an  incredible  decree  of  condescension  in  an  in- 

finite being',  finds  in  the  principle  above  stated  a  satisfactory  refutation.  As  compared 
with  infinitj ,  the  6  ]   irtion  of  matter  is  equal  to  the  greatest,  and  it  is  therefore  nc 

more  an  act  of  condescension  on  the  part  of  God  to  charge  himself  with  the  care  of  an  in- 
dividual than  of  a  nation — with  the  revolutions  of  a  satellite  than  with  the  movements  of 
a  system 


184  ALGEBRA. 

Dutch,  and  the  residue,  amounting  to  31,  Americans.     How  many  were 

there  in  the  whole  ? 

Ans.  120. 

(7)  Suppose  the  sound  of  a  bell  to  be  heard  at  the  distance  of  1142  feet  in 
a  second  in  a  still  atmosphere,  and  that  a  wind  is  blowing  sufficient  to  occa- 
sion a  delay  of  '  in  time.  In  how  many  seconds  will  the  sound  reach  a  dis- 
tance of  G000  feet  ? 

Ans.  6.304. 

(8)  Quicksilver  expands,  for  each  degree  of  the  centigrade  thermometer, 

s  .'.,,  of  its  volume.     According  to  this,  how  high  would  the  barometer  stand 

when  the  temperature  is  0°,  if,  when  tho  temperature  is  21°,  it  stands  at  a 

height  of  27  inches  .Q!  lines? 

Ans.  27  in.  7Ty38T  lines. 

(9)  What  degree  of  heat  in  a  centigrade,  thermometer  would  be  required 

to  cause  the  barometer  to  rise  to  26  inches  8  lines,  if  0°  raised  it  to  26  inches 

4  lines  ? 

Ans.  70ft. 

(10)  A  piece  of  silver,  the  specific  gravity  of  which  is  10J,  weighs  84  oz. 

How  much  weight  will  it  lose  in  water  ? 

Ans.  8  oz. 

(11)  In  a  mass  of  zinc  and  copper,  weighing  100  pounds,  8  parts  are  of  the 
former  and  3  of  tho  latter.  How  much  zinc  must  be  added,  that  the  propor- 
tions may  be  as  14 :5  ? 

Ans.  3 

(12)  At  the  extremities  of  two  arms  of  a  balanced  lever,  whose  lengths  are 
16  and  21  feet,  two  weights  are  suspended,  which  together  amount  to  65| 
pounds.     How  much  is  suspended  at  each  arm  ? 

Ans.  37^  and  28,      , 

(13)  The  rango  of  temperature  of  a  thermometer  during  the  year  was 
44T3ff°.  Tho  ratio  of  the  degrees  at  which  it  stood  at  the  extreme  points 
above  and  below  zero  was  7:4.     "What  were  the  points? 

Ans.  28fT'0-  above,  16385  below. 

(14)  In  4000  pounds  of  gunpowder  there  are  3210  less  of  sulphur  than  of 
charcoal  and  saltpetre,  2760  less  of  charcoal  than  of  sulphur  and  saltpetre. 
How  much  of  each  of  those  ? 

.  Sulphur  380,  charcoal  620,  saltpetre  3000. 

(15)  It  is  required  to  divide  the  number  99  into  five  such  parts  that  the  first 
may  exceed  tho  second  by  .">,  be  less  than  the  third  by  10,  greater  than  the 
fourth  by  9,  and  less  than  the  fifth  by  I 

\i.  .  The  i  arl  >  are  17,  1  '.  27,  8.  and  33. 

(if,)    \  ;:;,d  ]!  began  trade  with  eq  In  the  first  year  A  tripled 

his  stock,  and  bad    B27  to  Bpare;   B  doubled  his,  e    I  bad    £153  to  Bpare 

Now  the  amount  of  both  their  gains  was  five  times  the  stock  of  either.     What 

was  that  stock  ! 

Ans.  .£90. 

(17)    What  two  numbers  are  as  2  to  3  ;   to  each  of  which,  if  4  be  added,  tho 

Bums  will  be  as  .")  lo  7  ? 

Ans.  16  and  2  I. 

(181  Four  places  are  situated  in  the  order  of  the  letters  A    B.  ('.  D.    The 


SIMPLE  EQUATIONS.  185 

distance  from  A  to  D  is  34  miles.  The  distance  from  A  to  B  is  to  the  dis- 
tance from  C  to  1)  as  2  is  to  3  ;  and  one  fourth  of  the  distance  from  A  to  B, 
added  to  half  the  distance  from  C  to  D,  is  three  times  the  distance  from  B  to 
C.     What  are  the  respective  distances  '? 

Ans.  AB  =  12,  BC=4,  CD  =  18. 

(  I .')  A  field  of  wheat  and  oats,  which  contained  20  acres,  was  put  out  to  a 
laborer  to  reap  for  6  guineas  (of  21s.  each),  the  wheat  at  7  shillings  an  acre 
and  the  oats  at  5  shillings.  The  laborer,  falling  ill,  reaped  only  the  wheat. 
How  much  money  ought  ho  to  receive,  according  to  the  bargain  .' 

Ans.  <£4  lis. 


ad  only  half  his  army 
-}-G00  being  wounded, 
aither  slain,  taken  pris- 
isist  1 

Ans.  24000. 

.  party  of  soldiers,  who 
3k  from  him  j  of  what 
took  ^  of  what  now  re- 
leep  left.     How  many 

Ans.  103. 
;  the  expense  of  d€50  a 
by  i  of  what  remained 
k-  was  doubled.     What 


Ans.  740. 
ligits,  the  sum  of  these 
are  transposed.     What 

Ans.  23. 
mgers.  Seven  outsides 
The  fare  of  the  whole 
le  coach  took  up  3  more 
ce  of  which  the  faro  of 
to  15.     Required  the 


s,  18  and  10  shillings 


V-.J,    jli.o  »«»«  u,  c  ^~~..  ...~  .-0  J*  what  times  will  they  be 

together  during  the  next  12  hours  ? 

Ans.  "V,  minutes  past  1,  10}y«n'mutes  past  2,  and  so  on,  in  each  successive 
hour  5,5r  later. 

)  A  person  sets  out  from  a  certain  place,  and  goes  at  the  rate  of  11  miles 
in  ")  hours  ;  and  8  hours  after  another  person  sets  out  from  the  same  place, 
and  aoes  after  him  at  the  rate  of  13  miles  in  3  horns.  How  far  must  the  lat- 
ter  travel  to  overtake  the  former  ? 

Ans.  35J  miles. 

(27)  A  i-eservoir  which  is  full  of  water  may  bo  emptied  at  two  cocks.     One 
is  opened  and  J  of  the  water  runs  out ;  another  is  opened,  and  the  two  run- 


184  ALGEBRA. 

Dutch,  and  the  residue,  amounting  to  31,  Americans.     How  many  were 

there  in  the  whole  ? 

Ans.  120. 

(7)  Suppose  the  sound  of  a  bell  to  bo  heard  at  the  distance  of  11 12  feet  in 
a  second  in  a  still  atmosphere,  and  that  a  wind  is  blowing  sufficient  to  occa- 
sion a  delay  of  '  in  lime.  In  how  many  seconds  will  the  sound  reach  a  dis- 
'ance  of  G000  feet  ? 

Ans.  G.304. 

(8)  Quicksi  pands,  for  each  degree  of  the  centigrade  thcrmom 
-  '  .,  of  its  volume.     A 

when   the    temperature 
height  of  27  inches 

(9)  What  degi 

to  cause  the  baromctei 
4  lines  ? 

(10)  A  piece  of  silvc 
How  much  weight  wil 

(11)  In  a  mass  of  zit 
former  and  3  of  the  lat 
tions  may  be  as  14:5? 

(12)  At  the  extremit 
16  and  21  feet,  two  w 
pounds.     How  much  i^ 


(13)  The  range  of - 
44jV>.  The  ratio  of  i 
above  and  below  zero  w 


(11)  In  1000  pounds 
charcoal  and  saltpetre, 
How  much  of  each  of  i 

(15)  It  is  required  to 
may  exceed  the  -eecoiu 
fourth  by  9,  and  less  than  the  nttn  By  TV. 

\n  i.  The  parts  are  17,  14,  27,  8,  and  33. 

(In)  A  and  B  began  trade  with  equal  In  tin-  first  year  A  tripled 

hie   Stock,    and    had     627    to   spare;    B  doubled    his.    and    had  (£153    to   spare 

Now  the.  amount  of  both  their  gains  was  five  times  the  Bl  her.     What 

was  that  stock  ! 

Ans..  I 

(1?)  What  two  numbers  are  as  2  to  3;  to  each  of  which,  if  4  be  added,  the 

■urns  will  be  as  5  to  7  ? 

Ans.  in  and 
(18^  four  places  are  situated  in  the  order  of  the  letters  \    B.  C,  l>.     The 


SIMPLE  EaUATIO  185 

distance  from  A  to  D  is  31  miles.  The  distance  from  A  to  B  is  to  the  dis- 
tance from  C  to  D  as  2  is  to  3  ;  and  one  fourth  of  the  distance  from  A  to  B, 
added  to  half  the  distance  from  C  to  D,  is  three  times  the  distance  from  B  to 
C.     What  are  the  respective  distances  ? 

Ans.  AB  =  12,  BC  =  4,  CD  =  18. 
(  19)  A  field  of  wheat  and  oats,  which  contained  20  acres,  was  put  out  to  a 
laborer  to  reap  for  6  guineas  (of  21s.  each),  the  wheat  at  7  shillings  an  acre 
and  the  oats  at  5  shillings.     The  laborer,  falling  ill,  reaped  only  the  wheat. 
How  much  money  ought  he  to  receive,  according  to  the  bargain  ? 

Ans.  c£4  lis. 

(20)  A  general  having  lost  a  battle,  found  that  he  had  only  half  his  army 
-j-3600  men  left,  fit  for  action,  one  eighth  of  his  men  -j-GOO  being  wounded, 
and  the  rest,  which  were  one  fifth  of  the  whole  army,  either  slain,  taken  pris- 
oners, or  missing.     Of  how  many  men  did  his  army  consist  ? 

Ans.  24000. 

(21)  A  shepherd  in  time  of  war  was  plundered  by  a  party  of  soldiers,  who 
took  {  of  his  flock  and  ]  of  a  sheep  ;  another  party  took  from  him  J-  of  what 
lie  had  left,  and  \  of  a  sheep  more  ;  then  a  third  party  took  |  of  what  now  re- 
mained, and  \  a  sheep.  After  which  ho  had  but  25  sheep  left.  How  many 
had  he  at  first  ? 

Ans.  103. 

(22)  A  trader  maintained  himself  for  three  years  at  the  expense  of  <£50  a 
year,  and  in  each  of  those  years  augmented  his  stock  by  A  of  what  remained 
unexpended.  At  the  end  of  3  years  his  original  stock-  was  doubled.  What 
was  that  stock  1 

Ans.  740. 

(23)  There  is  a  certain  number  consisting  of  two  digits,  the  sum  of  these 
digits  is  5,  and  if  9  be  added  to  the  number,  the  digits  are  transposed.  What 
is  the  cumber  ? 

Ans.  23. 

(24)  A  coach  has  4  more  outside  than  inside  passengers.  Seven  outsides 
could  travel  at  2s.  less  expense  than  4  insides.  The  fare  of  the  whole 
amounted  to  <£0  ;  but  at  the  end  of  half  the  journey  the  coach  took  up  3  more 
oulside  and  one  more  inside  passenger,  in  consequence  of  which  the  fare:  of 
the  whole  became  increased  in  the  proportion  of  19  to  15.  Required  the 
number  of  passengers,  and  the  fare  of  each  kind. 

Ans.  5  inside,  9  outside  ;  fares,  18  and  10  shillings. 

(25)  The  hands  of  a  clock  are  together  at  12 :  at  what  times  will  they  be 
together  during  the  next  12  hours  ? 

Ans.  5,5,-  minutes  past  1,  lOAy-wiuutes  past  2,  and  so  on,  in  each  successive 
hou  r  5  r5T  later. 

I)  A  person  sets  out  from  a  certain  place,  and  goes  at  the  rate  of  11  miles 
in  5  hours;  and  8  hours  after  another  person  sets  out  from  the  same  place, 
and  goes  after  him  at  the  rate  of  13  miles  in  3  hours.  How  far  must  the  lat- 
ter travel  to  overtake  the  former  ? 

Ans.  35?  miles. 

(27)  A  reservoir  which  is  full  of  water  may  be  emptied  at  two  cocks.  One 
is  opened  and  ]-  of  the  water  runs  out ;  another  is  opened,  and  the  two  run- 


186  ALGEBRA. 

ning  together,  empty  the  vessel  in  \  c.f  an  hour  more  than  \vn<  require  1  fbi 
the  first  cock  alone  to  empty  the  fourth  part.  It'  the  two  cocks  had  been 
opened  at  the  commencement,  the  reservoir  would  have  been  emptied  in  \  of 
an  hour  sooner.     How  lung  would  it  have  tal  first  cock,  rimning'alone, 

ti)  empty  the  reservoir  1 

Ans.  4  hours. 

INDETERMINATE  ANALYSTS  OF  THE  FIRST  DEGREE. 

1  57.   If  there  be  proposed  for  solution  one.  equation  of  the  first  d 
taining  two  unknown  quantities,  any  value  at  pleasure  may  be  given  to  oi 
the  unknown  quantities,  and  tho  equation  will  make  known  a  corresponding 
value  for  the  other;  from  which  it  appears  that  the  equation  adm 
infinite  number  of  solutions.     The  number  of  solutions  will,  however,  not  bo 
so  unlimited,  if  it  be  required  that  the  values  of  x  ami  y  shall  be  whole  num- 
ber.- :   and  still  less  so,  if  they  must  be  both  entire  and  positive. 

Let  there  be  the  equation 

ax-\-by=c, 
a,  b,  c  being  any  whole  numbers  whatever,  either  positive  or  ne: 
all  the  factors  common  to  these  three  numbers  could  be  su] 
this  to  have  been  done. 

And  first,  let  it  be  observed,  that  if  there  should  remain  now  a  commin  lac- 
tor  in  a  and  b,  the  equation  could  not  admit  of  a  solution  in  whole  numl 
for  whatever  values  might  be  substituted  for  x  and  //,  the  first  mi  would 

be  divisible  by  this  common  factor  of  a  and  Z>,  while  the  second  m  ivould 

not,  and  the  equality  would  therefore  be  impossible:  a  and  b  must,  therefore 
be  supposed  prime  to  each  other. 

.  Take,  for  example,  the  equation 

24jr+65//=243 "  .  .  .  (1) 

in  which  the  coefficients  24  and  65  are  prime  to  each  other. 

Resolving  it,  with  respect  to  x, 

243—1  3  — 17  v 

rr= -  =  10 °?/4- 

24  W      ~^+      24      ' 

In  order  that  x  and  y  may  both  bo  whole  numbers,  and.  at  the  - 

satisfy  tho  given   equation,  it  is  necessaiy  that  — — —  should  be  a  : 

nun.! 
Representing  this  by  /,  we  have 

3  — 17// 


and 


=  «     (*) 


.,—10  —  2//  +  / (3) 

i  of  the  giveYi  equation  in  whole  numbers  then  reduces  itself  to 
i  f  the  equation  (2). 
We  resolved  the  given  equation  with  re  peel  to  the  unknown  quantity  which 
had  tli*'  least  co  ;  doing  the 

:;  — 

3F-        — «+  ir-; 

nnd  procee   ing  as  bei 


JNj  .'IK  ANALYSIS  OF  THE  EE. 

3— 7< 

=*' (4) 

17  V   ; 

y=—t  +  L' (5) 

The  solni ion  of  (2)  in  whole  numbers  depends  on  that  of  (I,  .  re- 

o*ct  to  /,  gives 

•   3  — 17 1'  3  —  31' 

«— j ■-=<•+— 

'-=¥='•  • (<9 

t=— 2i'+r (7) 

the  same  way, 

3—7/"  r 

/' 1  Of" — 

t-     3     _i     a       3 
■o=«'" (8) 

f=i— sr— r" (9) 

Equation  (8)  gives 

/"  —  ?,','" (10) 

The  solutions  of  the  given  equation  in  whole  numbers  are  therefore  obtained 
by  giving  to  the  indeterminate  quantity  V"  all  possible  values  in  whole  mini 
bers,  positive  or  negative;  and  for  each  of  these  values  of  I'",  the  i 
(10),  (9),  (7),  (5),  and  (3),  determine  successively  the  values  of  the  indeter 
minate  quantities  t",  (',  t,  and  of  the  unknown  quantities  y  and  x.     The  equa- 
tion is  therefore  resolved  in  the  [uired. 

Formulas  may  be  obtained  which  give  immediately  the  values  of  .r  and  y  in 
terms  of  V".  For,  substitutim  he  value  31'"  of  t"  in  (9),  we  find*'=l  —  1'"' \ 
substituting  this  value  of  t'  and  that  of  t"  in  (7),  we  find  t=  —  2  +  17t'' ;  sub- 
stituting this  last  value  and  that  of  V  in  equation  (5),  we  find  y=3— 2it'",  and 
from  (3),  x=2  +  6ol'". 

These  last  two  expressions  give  all  the  entire  solutions  of  the  proposed 
equations  by  attributing  successively  to  t'"  all  possible  values  in  entire  num- 
bers, positive  or  negative. 

159.  The  same  process  with  the  general  form 

ax-\-by=c 
would  run  thus, 

c — ax 

y=-v- w 

Dividing  a  by  /;,  aud  calling  q  tho  quotient,  r  the  remainder, 

c  —  (bq-\-r)r  c — rx 

y  = p =-?*+-£-, 

mak  e 

c—rx  c—bt 

— =<•  ■■■  *=>— <2> 

railing  «j '  the  quotient  of  b  by  r,  and  r'  the  remainder, 

c—r't 
x——q>t+—-—, 


=«',.-.  <=--7-.   •     '     •   (3) 


r  r 


198  ALGEBRA. 

And  calling  q"  the  quotient  of  r  by  r',  and  r"  the  remainder, 

c—r"t' 
t=-q"t'+-^—, 


make 

c—r"V 


(-1) 


r 

and  so  on.     The  process  is  now  evident,  and  it  will  be  perceived  Hint  the  co- 
efficierjt8  r,  r',  ?",  which  enter  into  the  equations  (2),  (3),  (4),  are  the  suc- 
cessive r  -;  which  would  be  obtained  in  operating  as  if  to  find  the  com 
ition  divisor  of  a  and  b.     We  must  at  length  arrive  at  a  remainder  1,  be. 
a  Hi  id  b  are  supposed  prime  to  each  other. 

For  the  sake  of  being  more  definite,  let  r"  be  supposed  to  be  this  remaindei 
then  equation  (4)  gives 

t'=—  r'l"  +  c (5) 

By  means  of  equations  (2),  (3),  (4),  and  (5),  the  values  of  y,  x,  I,  and  V  idh 
be  written  as  follows  : 

y=—(]z    +t 
xz=—q't    -\-t' 
t  =—q"t'  +  l" 
t'  =  —r'l"  +c. 
T  of  equations  shows  that  any  entire  value  bein°;  assumed  for  t". 

Lng  value  of  V  substituted  in  that  of  t,  tho  values  of  t,  tf  in  tlial  of  r,  and 
the  \  allies  of  x,  t  in  that  of?/,  the  proposed  equation  is  resolved  in  whole  nui 

Tho  success  of  tho  method  is  founded  on  the  pro  .lion 

which  division  effects  upon  the  coefficients  of  the  indeter 
reason,  however,  why  the  constant  term,  found  in  the  successh  dons, 

should  not  also  be  divided.     In  this  way  tho  calculation  will  it  nailer 

numbers,  an  advantage  which  is  not  to  be  neglected. 
For  example,  take  tho  equation 

3x— 8?/ =  4  3. 
^s  the  multiplier  of  x  is  less  than  that  of?/,  resolve  the  equation  .Ter- 

ence to  x, 

8y+43 
s=— — . 

Dividing  8  by  3,  the  quotient  is  2,  and  the  remainder  2 ;  and  dividing  43  ov 
.",  the  quotient  is  14,  remainder  1;  then 

2//+1 
ar=22/  + 1  '*  +  ''-J-  =  '.'/  + 1 1  + ' 

2t/+l=3/ 

3l  —  \  t  —  \ 

y=-^-=t+—=t+t' 

l  —  l=2t' 

in  which  last  equality  V  may  receive  all  possible  entire  values.     By  means  oi 
this  value  may  be  found 

v=t+t'=2/'+i+r=r>t'+\ 

i=2y+14+«=2(3f,+l)+14+2f+l=8«'+17. 

to  V  the  values  0,  1,  2,  3,...  we  lind 


INDETERMINATE  ANALYSIS  OF  THE  FIRST  DEGREE.  133 

y=   1,    4,    7,  10,... 
*     .r=17,  25,  33,  41,... 
V  may  also  receive  the  negative  valu 

1    o    T 

161.  In  the  above  example,  the  values  of?/  and  x  form  two  arithmetical  pro- 
gressions, the  first  of  which  has  the  common  difference  3,  the  coefficient  of  x 
in  the  proposed  equation  ;  and  the  second  the  common  difference  8,  the  co- 
efficient of  y  taken  with  the  contrary  sign.  This  proposition  may  be  sei  i  to 
be  goneral  by  elfecting  the  successive  substitutions  in  the  general  solution. 
but  the  following  demonstration  is  preferable. 

It  appears,  from  the  general  investigation  already  made,  that  the  equation 

az~\-by=c (1) 

admits  of  an  infinite  number  of  solutions  in  whole  numbers,  whatever  may  be 
the  signs  of  a  and  b,  provided  they  are  prime  to  each  other.  Suppose  one  of 
these  solutions  to  be 

x=A,  y=B. 
These  values  must  satisfy  the  given  equation  (1),  thus, 

«A-foB=c. 
Subtracting  this  equality  from  (1),  we  have 

a(x— A)-f%  — B)=0 
_       a(A — x) 

The  values  of  x  are  to  be  whole  numbers,  and  such  that  y  shall  also  be  a 
whole  number.  Then  the  product  a(A — x)  must  be  divisible  by  b ;  but  a  is 
prime  with  b,  (A — x)  is,  therefore,  a  multiplier  of  b  (see  Art.  84,  Note),  hence 
we  may  write 

A— x=bt; 
t  being  somo  whole  number.     From  whence 

.r=A  —  bt,  y=B-\-at. 

These  formulas  exhibit  the  law  of  the  values  to  be  obtained  for  x  and  y, 
when  there  are  given  to  t  all  entire  values  successively.  If  t  be  taken  equal 
to  0,  1,  2,  3,  . ...  there  results 

x=\,  A— b,  A— 26,  A— 36,  &c. 
y=B,  B-f  a,  B  +  2a,  B  +  3a,  &c. 

In  general,  when  t  increases  by  unity,  y  increases  by  a,  and  x  by  --6. 
The  solutions  in  whole  numbers,  then,  of  the  equation  ax+by=c,  are  the  cor 
responding  terms  of  trco  progressions  by  differences.  In  the  progression  be- 
longing to  each  of  the  indeterminates,  x  and  y,  the  common  difference  is  equal  to 
the  coefficient  of  the  other  indeterminate.  But  it  is  necessary  to  be  careful  to 
take  one  of  the  coefficients  with  the  same  sign  that  it  has  in  the  equation,  and 
(he  other  with  the  contrary  sign. 

It  is  immaterial  which  of  the  coefficients  is  taken  with  the  contrary  sign, 
because  in  the  formulas  which  express  x  and  y  the  signs  of  bt  and  — at  may 
be  changed,  since  t  can  receive  all  possible  values,  positive  and  negative. 

162.  In  the  general  equation,  if  c=0,  so  that 

ax-\-by  =  0, 
as  one  solution  is  eviden.y  x=Q,  y=0,  the  general  formulas  become 

x—bt,  y= — at. 


I'M  ALQE] 

iG3.  Again,  suppose  c  to  be  a  multiple  of  a  or  b.     Let  c=bd,  then 

-j-/w/  =  /;cZ. 

One  solution  is  evidently  r=0,  //  =  </  ;  hence  the  general  values  are 

x—bt,  y=d  —  at. 
Example,  5.r  — 7//=21. 

The  evident  solution  is  x=0,  y=—'3,  and  the  general  values 

x=7t,y=—3-\-r,/. 

164.  We  shall  point  out  two  simplifications  which  may  sometimes  be  made 
uj  the  calculations.     An  example  will  explain  them. 

80ar-yl77/=39. 
lolving  it  with  respect  to  y, 

80a:—: 

If  80  be  divided  by  17,  80  =  17  X  4  + 12  ;  but  as  the  remainder,  12,  exceeds 

half  the  divisor,  17,  we  observe  that  we  may  write 

80  =  17  X  (4  +  1)  + 12  —  17  =  17X5— 5; 

that  is,  augmenting  the  quotient  by  unity,  we  have  a  negative  remainder  less 

than  half  the  divisor,  which  causes  a  more  rapid  reduction  in  the  numbe 

The  39,  divided  by  17,  leaves  a  remainder  -\-o,  which  it  is  unnecessary  to 

change.     "\\  e  have  then 

(17  X -3  —  5)x—  17X2  —  5  5.r-f5 

y= rj =5x-2— — . 

But  another  simplification  now  ]  itself,  from  the  fact  that  5  is  a  factoi 

of  5.r-[-5,  and  this  numerator  may  be  written  5(x-J-l).  In  order  to  rendei 
5(x-\-l)  divisible  by  17,  it  is  011I3-  necessary  to  take  x-\-l,  any  multiple  what- 
ever of  17.     Whence  the  auxiliary  equation 

.r -4-1  =  17/; 
.-.  .(-=17/  —  l.-y  =  S0t— 7. 

RESOLUTION    OF   THE  EQUATION   C.C-\-l,y  =  C  IN   NUMBERS    EOTII    ENTIRE  AND 

POSH  !VK. 

165.  We  begin  as  if  the  values  of  x  and  y  were  required  to  be  entire  only, 
and  thus  derive,  as  before,  e  ons  of  the  form 

x=\—  bt,  y=B  +  at. 
But  now,  instead  of  attributing  to  /  all  possible  values  in  whole  numbers,  we 
choose  only  those  which  will  render  x  and  y  positive.     Hence  there  result  for 

t  certain  limitations  Which  are  always  easy  tO  determine. 

First,  let  us  consider  the  case  where  a  and  b  have  the  same  sign  in  the 

equation 

ax-\-by=c (1) 

Suppose  a  and  b  positive;  for  if  they  were  both  negative,  they  might  be 

rendered  positive  by  changing  all  the  signs  of  tin'  equation.     We  must  also 

ppose  c  to  be  positive,  otherwise  the  equation  would  be  impossible  in  p©si- 

whole  numbers. 
Writo  the  general  values  of.c  and  y  under  the  following  form: 

tel(£_i)i3^«(«_z2). 

Then  we  perceive  that,  to  render  x  positive,  it  i 

A                                                                                                                                                                                                                  .-I'- 
ll! take  t<C  ,  ,  and  likewise,  in  order  that   7  may  be  positive,  to  tako  /> . 


V 

INDETERMINATE  ANALYSIS  OF  THE  FIRST  DEGEE  i91 

The  signs  >  and  <  do  not  exclude  equality  ;  that  is  to  say,  if  the  first  limit 
were  a  number  n,  we  might  make  i  =  /i.  The  corresponding  value  of  x  would 
be  r=0. 

166.  Since  t  must  be  an  entire  number  between  two  limits,  it  follows  that 
the  number  of  solutions  of  the  equation  is  also  limited. 

And  this  is  evident  from  the  equation  itself;  for  a  and  b  being  positive   if 
ilute  for  x  and  y  positive  numbers,  the  two  terms  ax-\-by  will  be  al- 
i  positive ;  and  as  their  sum  has  to  remain  constantly  equal  to  c,  it  is  im- 
possible that  either  of  these  terms  should  increase  indefinitely. 

It  may  happen  that  there  is  no  whole  number  between  the  limits  assigned 
above  for  t ;  then  we  conclude  that  the  equation  is  impossible.  Such  a  case 
would  happen  if  the  limits  should  be  embraced  between  two  consecutive  whole 
numbers  like  these,  <>4.'t  and  £<4f ;  or,  again,  if  they  were  contradictory, 
as,  for  example,  <>4g  and  £<3f. 

1G7.  In  the  second  place,  consider  the  case  in  which  a  and  b  are  of  contrary 
signs.     Suppose  the  equation  in  question  to  be 

ax — by=c (2) 

in  which  a  and  b  represent  two  positive  numbers.     Then  the  general  v. 
of  x  and  y  are  of  the  form 

x=A-{-bt,  y  =  B-\-at. 

But  we  can  write  them 

x=b{t-~-),y=a{t-=^). 

And  wo  perceive  at  once  that  to  have  x  and  y  positive,  we  must  have,  at  tho 
same  time, 

that  is  to  say,  we  may  attribute  to  t  all  entire  values  above  the  greatest  of 
these  limits  without  excluding  equality,  if  this  limit  is  an  entire  number. 

By  this  wo  perceive  that  the  equation  ax — b>j=c  admits  always  of  an  infinite 
number  of  solutions,  while  the  equation  ax-\-byz=c  admits  of  but  a  limited 
number,  and  even  may  not  have  any. 

Let  us  apply  what  precedes  to  some  problems. 

168.  Problem  I. — A  company  of  men  and  women  expend  at  a  feast  1000 
francs.     The  men  pay  each  19  francs,  and  the  women  11  francs.     How  - 

n  and  hoiv  many  ivomen  arc  there? 

Let  x  represent  the  number  of  men  and  y  the  number  of  women.  "Wo 
have  to  resolve  in  entire  numbers  the  equation 

19.r+lly=1000 (3) 

In  making  the  calculation,  as  in  (160),  and  profiting  by  the  simplifications  in- 
dicated by  (Art.  164),  we  have  successively, 

1000  — 19.r  •  :•>'•— 1 

y= =  91-2.r+-ir-=91--r+; 

3.r  —  \=\\t 
11*4-1  \—l 

l—t=3t' 
t=l—3t'. 
Arrived  at  this  point,  we  return  to  x  and  y,  and  they  become 


1 92  ALGEBRA. 

x=it+t'=m— 3i')+t'=i— a  f 

2/  =  'Jl—  2.r+/  =  91  —  2(4  —  11/')  + (1  —  3f')  =  8-i-r19t'. 
Thus,  llie  general  formulas  which  express  .r  iiuJ  ?/  in  terms  off  are 

x=4  — 11/',  ?/  =  84  +  19/'. 
In  order  that  a:  may  be  positive,  it  is  necessary  and  sufficient  that  we  ham 
11/' <4,  or  £'<CfY  ;  and  in  order  that  y  should  be  also  positive,  it  is  necessary 
and  sufficient  that  we  have  19f  >  — 84,  or  /'>  — 4,^.     Then  wo  must  take  /'. 
one  of  the  series  of  values, 

t'  =  0,  —1,  —2,  —3,  —4. 

To  these  values  correspond 

a=4,    15,  26,  37,  48 
?/=84,  65,  46,  27,     8. 
The  number  of  solutions  is  limited,  as  we  ought  to  expect,  since,  in  the 
equation  (3),  the  terms  containing  x  and  y  are  of  the  same  sign. 
There  are  five  solutions  in  all,  to  wit ; 

1st  solution,  4  men  and  84  women. 
2d  solution,  15  men  and  65  women. 
3d  solution,  26  men  and  46  women. 
4th  solution,  37  men  and  27  women. 
5th  solution,  48  men  and    8  women. 

Remark. — From  what  has  been  said  at  (161),  it  is  sufficient  to  procure  a 
single  solution  of  the  equation  (3)  to  form  immediately  the  general  values  of  x 
and  y.  Thus,  after  having  found  above  t=l—  3/',  we  make  t'  =  0  ;  and  if  we 
calculate  the  corresponding  values  t  =  l,  x=4,  i/  =  84,  it  is  evident  that  the 
values  x=4,  y=84,  ought  to  form  one  solution  of  the  equation  ;  then  we  i 
place  immediately  x=4  — 11/',  ?/  =  84  +  19/'. 

169.  Problem  II. —  With  two  measuring  rods  of  different  lengths,  the  one  5 
feet,  and  the  other  7,  it  is  required  to  make,  by  placing  them  the  one  after  tht 
other,  a  length  of  '23  feet. 

This  problem  requires  tho  solution  in  whole  numbers  of  the  equation 

5x+7t/=23. 

Wo  derive  from  it  successively 

23—71/  2+2y 

x=—^-=5-y--^=o-y-2t 

1  +  2/  =5t 
y=5t-l 
x=zC>  —71 

In  order  that  y  may  bo  positive,  we  must  make  />'  ;  and  that  x  may  be 
positive,  /<','.  As  no  whole  number  falls  between  ',  and  f,  wo  conclude  that 
the  problem  is  impossible. 

Ki  makk. — The  equation  would  have  had  an  infinite  number  of  solutions  if 
negative  values  had  been  admitted.  For  example,  if  /=0,  we  hue  .r=6, 
i/r=  —  1.     This  solution  indicates  thai  byplacii     one  of  the  rods,  that  of  5  feet, 

6  limes  in  succession,  and  placing  afterward  the  rO  1  of  7  l>  cut  off 

iis  length  from  the  end  of  the  distance  thus  ol  tained,  the  remainder  would  be 
the  required  length,  23  feet 

.    PaoBLl  M    [II.        I    /rrson  pi  '.iircs  an*/  sheep. 

ha-'-  cost  him  8  shilling*,  and  each  thru'  27.     Jh  found  Uiat  he  had  paid  Jet 


INDETERMINATE  ANALYSIS  OF  THE  FIRST  DEGREE.  193 

the  hares  97  shillings  more  than  for  the  sheep.     How  many  hares  did  he  pur 
chase,  and  how  many  sheep  ? 

8x—27y=97 
27w+97  3y+l 

3y+l=8t 

8t—l  t+l 

y=-ir=3t--jr=3t-l' 

<+l=3f 
t=3t'  —  l. 
By  making  ('=0,  we  have  t=  —  1,  y  =  — 3,  2=2.     And  the  general  values 
are 

T=27f+2,  yz=Sl'  —  3. 
The  values  of  x  and  y  having  to  be  positive,  these  formulas  show  that  t 
ought  also  to  be  positive,  and  large  enough  to  cause  8£'>3,  or  i'>  \.     We  may 
then  give  to  V  all  the  values  $'— 1,  2,  3,  &c,  to  infinity  ;  and  we  form,  conse- 
quently, the  table, 

t'  =   1,    2,    3,      4,  &c. 

:r=29,  56,  83,  110,  &c. 

y=  5,  13,  21,    29,  &c. 

The  problem  admits  of  an  infinite  number  of  solutions ;  and  the  answer  is, 

that  there  are  29  hares  and  5  sheep,  or  56  hares  and  13  sheep,  or  83  hares 

and  21  sheep,  &c. 

171.  Problem  IV. — To  find  a  number  such  that,  in  dividing  it  by  11,  there 
remains  3,  and  dividing  it  by  17,  there  remains  10. 
Let  the  number  be  represented  by  N,  then 

N  =  ll.r+3  and  N  =  17i/+10 

.-.  ll.r+  3  =  17v/+10 (6) 

Proceeding  as  before, 

17.V+7         .  fy-l-7 

x=-jr-=y+—1-=y+t 

6y  +  7  =  Ut 
Ut  — 7  t+l 

y  =  -^  =  2l-l-^  =  2t-l-f 

t-\-l=6t' 
t—6t'  —  l. 
The  hypothesis  ('=0  gives  t=  —  1,  ?/=  —3,  x=  —4  ;  and  then  we  conclude 
immediately  that 

x=17l'— 4,  y=lll'— 3. 

We  can  not  take  V  negative,  nor  even  t'=0,  because  x  and  y  would  become 
negative;   but  we  may  take  t'r=l,  2,  3,  &c.,  to  infinity. 

If  we  wish  formulas  in  which  Ave  can  give  to  the  indeterminate  all  entire 
positive  values  setting  out  from  zero,  all  that  is  necessaiy  is  to  chauge  t  into 
1+0,  6  being  the  new  indeterminate.     Then  we  have 

i=13+ 170,  y=8+U0. 
By  means  of  these  values,  we  find 

N=llx+   3  =  11(13  +  170)+   3  =  146+1870 
N  =  17?/+10  =  17(  8+110)  +  lO  =  146+1870. 
These  two  expressions  are  equal,  and  they  should  be,  since  equation  (6)  has 

N 


J  94  ALGEBRA. 

been  formed  by  equating  the  values  of  N.  We  perceive  that  there  19  an  in- 
finity of  numbers  which  fulfill  the  two  conditions  enunciated,  and  that  they  are 
all  represented  by  the  formula 

N  =  146+1870, 
in  which  6  is  an  indeterminate,  which  may  receive  all  positive  values  beginning 
with  zero. 

It  is  easy  to  6how  that  this  number  N  satisfies  the  enunciation ;  that  is  U 
say,  that  if  we  divide  it  by  11,  the  remainder  will  be  3,  and  if  by  17,  the  re- 
mainder will  be  10  ;  for  wo  have 

N  3  N  10 

n=170+13  +  n,  and  -=n0+8+-. 

172.  Problem  V. — To  find  a  number  such  Qiat,  dividing  it  by  11,  there 
remains  3  ;  dividing  by  17,  there  remains  10  ;  and  dividing  it  by  37,  there  re- 
mains 13. 

In  the  preceding  problem  we  have  found  the  numbers  which  fulfill  the 

first  two  conditions.     Putting  x  for  0,  which  we  may  do,  since  6  can  be  any 

positive  whole  number,  this  formula  becomes 

N=146+187.r (8) 

But  in  order  that  the  number  N  may  fulfill  the  third  condition,  we  must 

have  N=:37y-J[-13.     Then  we  have  the  equation 

37y+13  =  146-r-187;r. 

Then 

1872T+133  2i+22 

V= ^ =5z+3+— ^-=5*+3+2i 

x+ll=37f 
xz=37t— 11. 

In  order  that  x  may  be  positive,  we  must  give  to  t  only  positive  values  above 
zero.  But  in  making  t=\-\-6,  wo  can  attribute  to  8  all  the  entire  positive 
values  beginning  by  zero.     By  this  change  x  becomes 

x=26  +  370. 
And  by  substituting  this  value  in  formula  (8),  we  obtain 

N=5008+ 69190. 
Such  is  the  general  formula  of  the  numbers  which  satisfy  the  three  condi 
tions  enunciated. 

173.  The  determination  of  tho  limits  led  to  the  necessity  of  finding  (165) 
the  values  of  tho  final  indeterminate  t,  which  render  positive  expressions  of 
the  form  A-\-bt,  or,  in  other  terms,  which  are  such  as  to  make 

A+6t>0. 
Transposing  tho  term  A, 

&*>—  A. 
If  b  is  positive,  dividing  by  b, 

But  if  b  is  negativo,  tho  division  by  b  changes  the  signs  of  the  inequality 
and  tho  two  members  are  unequal  in  the  contrary  sense  ;  i.  e., 

«4 

Suppose,  more  generally,  that  we  have  ilu>  inequality 

at+b>ct+d. 


INDETERMINATE  ANALYSIS  OP  THE  FIRST  DEGREE.  195 

By  the  transposition  of  the  terras, 

(a — c)t^>d — b. 

Then,  according  as  a — c  is  a  positive  or  negative  quantity,  we  derive 

d-b            d-b 
£> ,  or  /< . 

a — c  a  —  c 

This  process  is  called  resolution  of  inequalities.     The  whole  subject  cf  in 
equalities  will  bo  found  treated  in  a  subsequent  article. 

174.  Resolution  in  whole  numbers  of  several  equations  of  thk 
first  degree,  when  the  number  of  equations  is  less  than  that 
of  the  unknown  quantities. 

Let  there  be  for  resolution  the  equations 

2x+Uy— 7z=341 (1) 

10.r+   4?/  +  9c  =  473 (2) 

If  we  multiply  the  first  equation  by  5,  and  afterward  subtract  the  second, 
we  shall  have 

66y  — 44z=1232. 
Or,  dividing  by  22, 

Zy— 2z=56 (3) 

But  the  entire  values  of  y  and  z,  which  suit  the  proposed  equations,  ought 
also  to  satisfy  this  ;  consequently,  applying  to  it  the  method  already  known, 
we  have 

y=2t,  z~Zt— 28. 
If  we  had  but  equation  (3),  we  should  have  its  solutions  in  whole  numbers, 
by  giving  to  t  all  the  whole-number  values  possible.  But  this  equation  takes 
the  place  of  only  one  of  the  proposed,  so  that  it  is  necessary  that  the  values 
of  y  and  z  should  be  such  that,  in  adding  to  them  certain  values  of  x,  which 
must  also  be  entire,  one  of  these  proposed  equations  shall  be  verified.  For 
this  reason  we  substitute  the  preceding  values  of  y  and  z  in  equation  (1),  and 
seek  for  the  entire  values  of  x  and  I,  which  belong  to  the  resulting  equation 
The  substitution  gives 

2.r+ 7f=145; 
and  from  this  we  obtain,  designating  by  V  any  whole  number  whatever, 

.T=69  +  7i',  t  =  l—2t'. 
Then  place  the  value  t=\  —  2t'  in  those  of  y  and  z,  and  you  find  the  un- 
known quantities  x,  y,  z  expressed  in  terms  of  t\  to  wit : 
x=69+7t',  y=2— W,  z  =  —  25— 61'. 
These  formulas  make  known  all  the  entire  values  which  satisfy  the  equa- 
tions proposed. 

If  it  be  desired  besides  that  these  values  should  be  positive,  t  must  be  so 
chosen  that 

69+7*'>0,  whence  f>— 9«; 
2— 4i'>0,  whence  f<       A; 
— 25  —  6r>0,  whence  «'<—  4£. 
From  this  we  find  the  only  values  which  can  be  attributed  to  V  are  t'z=z — 5, 
—  6,  — 7,  — 8,  — 9.     By  substituting  these  numbers,  we  6hall  have  five  solu- 
tions in  positive  wholo  numbers  : 

z=34,  27,  20,  13,  6 
i/=22,  26,  30,  34,  38 
z=  5,  11,  17,  23,  29. 


19b  ALGEBRA. 

175    The  p  g  example  shows  sufficiently  the  method  to  be  pun      ! 

in  resolving  equations  of  the  first  degree  in  positive  whole  numbers,  when  t lie 
number  of  unknown  quantities  e  that  of  tho  equations.     But,  to  leave 

nothing  to  be  desired,  1  shall  indicate  the  method  to  be  pursued  in  the  case 
of  three  equations. 

Let  there  be,  then,  '  in  the  unknowns  x,  y,  z,  u  three  equations  of  the 

1st  degree,  which  I  will  name  collectively  the  equations  [A]. 

By  tho  elimination  of  x  we  shall  find  between  y,  z,  and  u  two  equations  of 
the  1st  degree  :  I  shall  name  them  [B]. 

By  the  elimination  of  y  we  shall  deduce  from  these  last  an  equation  of  the 
1st  degree  between  ;  and  it."   1  shall  name  it  [C]. 

From  tho  equation  [C]  wo  derive  z  and  u  expressed  in  function  of  an  aux 
diary  indeterminate  t. 

These  values  being  substituted  in  oue  of  the  equations  [B],  we  derive  from 
it  an  equation  between  y  and  t,  and  from  this  tho  values  of  y  and  (  in  function 
of  a  now  indeterminate  «'  ;  consequently,  we  can  also  express  z  and  u  in  terms 

ofr. 

Finally,  these  values  of  y,  z,  u  being  carried  into  one  of  the  equations  [A], 
there  will  result  an  equation  between  x  and  V,  which  will  enable  us  to  find  x 
and  I',  and,  consequently,  y,  z,  and  it,  in  function  of  a  new  indeterminate  /". 

When  the  equation  is  to  be  resolved  in  whole  numbers  of  any  sign  what- 
ever, we  may  attribute  to  tho  final  indeterminate  L"  all  possible  valui 
whole  numbers.     But  when  the  solutions  are  to  be  restricted  to  such  as  are 
at  the  same  time  entire  and  positive,  there  will  exist  for  L"  limitations  which  it 
will  be  always  easy  to  assign. 

17G.  When  we  have  two  more  unknowns  than  equations,  or  several  more, 
the  ^determination  is  still  greater  ;  but  the  condition  of  having  values  which 
shall  be  at  the  same  time  entire  and  positive,  may  limit  considerably  the  num- 
ber of  solutions.  Wo  shall  confine  ourselves  to  two  examples,  which  will  suf- 
fice to  show  how  the  method  explained  above  should  be  modified  in  such  c 

Given  to  resolve  in  positive  whole  numbers  the  equation 

10x+9?/+7r=5S (4) 

As  the  unknown  z  has  the  smallest  coefficient,  I  derive 

58— 9?/— 10.r 

;= 1 — ; 

and,  effecting  the  division  as  far  as  possible, 

2— Oy— 3.r 
z=8-y-z+ •- . 

Tho  numerator  2 — 2y — 3x  must  bo  a  whole  numb  ;  theio 

fore  I  place 

?  —  2  /  —  ■:>r  =  7l  : 
o_;}.r_7,  x+t 

■■■li= r2 =l—r-r,/_  -J-; 

and,  x-^-t  being  obliged  to  bo  a  whole  number  divisible  by  2,  I  place,  also, 

z+t=2?  .-.  r=—  t+2tf ; 
and,  going  hack  to  y  and  c,  we  expr<  unknowns  in  function  of  t  and  t 

We  have  thus  the  three  formulas 

*=— t+2f,  y=sl— 2f— 3C,  :=7+l/-f/'  ....  (5) 


INDETERMINATE  ANALYSIS  OV  THE  FIRST  DEGREE.  19? 

In  oiiler  to  have  the  entire  and  positive  solutions  of  the  proposed  equation 
(4),  wo  must  give  to  t  and  t'  all  the  entire  values,  which  satisfy  simultaneously 
the  three  conditions 

—  l+2t'>0,  l—2t— 3f  >0,  7  +  4H-«'>0  ....  (6) 
From  hence  result  limitations  for  t  and  V,  which  will  be  discovered  by  em 
ploying  for  these  inequalities  operations  altogether  analogous  to  those  of  elimi- 
nation.    For  greater  neatness,  suppose  the  signs  >  exclude  equality  ;  that  is 
to  say,  that  hone  of  the  three  unknowns,  x,  y,  and  z,  can  be  zero. 
First,  if  we  multiply  the  1st  by  3  and  the  2d  by  2,  they  become 
_3«+6«'>0,  2  —  it— 6£'>0  ; 
adding,  t'  disappears,  and  we  have 

2— 7«>0  .'.i.<|. 
A  similar  elimination  between  the  second  inequality  and  the  third  gives 

22  +  10«>0.-.  <>—  2i. 
We  see  that  the  indeterminate  t  is  embraced  between  the  limits  — 2]  anu 
f  ;  then  we  shoidd  take  only 

t=-2,  -1,0. 
Let  us  consider  each  of  these  values  successively. 
1°.  If  we  make  t= — 2  in  the  three  inequalities  (6),  they  become 
2+2«'>0,  5— 3f>0,  —  l+f>0; 
.-.«'> -1,«'<1|,  f>l. 
As  there  is  no  whole  number  between  1  and  1|,  it  follows  that  the  value 
t= — 2,  which  furnishes  these  limits  for  t',  ought  to  be  rejected. 
2°.  If  we  make  t=  —  1,  the  three  inequalities  (6)  become 
l  +  2«'>0,  3— 3«'>0,  3  +  *'>0  ; 

.-.f>— |.  r<+i.tf>— 3. 

Between  — \  and  -f-1  there  is  no  other  entire  number  except  0  ;  then  we 
can  take  t=  —  1  and  t'  —  O. 

3°.  If  we  make  <=0,  the  inequalities  become 

2i'>0,  1— 3f>0,  7-K>0; 
.:(.'> 0,  V<l,V>-7. 
Between  0  and  1  there  is  no  whole  number ;  consequently,  the  value  i=0 
ought  also  to  be  rejected. 

The  only  values  of  t  and  V  to  which  positive  values  in  whole  numbers  of  x, 
y,  and  z  correspond  are,  then,  t-=  —  1  and  £'=0.  By  substituting  them  in 
the  formulas  (5),  we  obtain 

x=l,  7/ =3,  2=3, 
and  this  solution  is  the  only  one  admissible. 

177.  For  a  second  example,  I  propose  the  two  equations 

6.r+   77/+3r+2*<  =  100 

24x+12y4-7z+3tt=200. 

Eliminating  v,  we  have 

3O.r+3?/+5r  =  100. 

As  in  this  equation  the  terms  30.r  and  100  are  divisible  by  5,  it  will  be  best 

to  take  the  value  of  z  :  this  is 

3?/ 
z=20  —  Gx  —  -~. 
5 

From  which  we  see  that  y  ought  to  be  a  multiple  of  5  ;  consequently,  we  have 

y=5t 

z=20— 6x— 3«; 


198 


ALGEB1IA. 


then,  by  substituting  these  values  in  the  first  of  the  two  proposed  equation* 
it  becomes 

G.C+35J+G0  —  18x— 9t+2u  =  100 ; 
or,  rather, 

— 12x+26J+2u=40; 
.-.  u=20  +6z— 13  . 
The  three  unknowns,  y,  z,  u,  are  thus  found  expressed  in  functions  oi  2, 
and  of  the  indeterminate  auxiliary  t. 

In  order  to  resolve  the  two  proposed  equations  in  positive  numbers,  it  is  evi- 
dently necessary  to  take  x  and  t  positive,  since  x  is  one  of  the  primitive  un- 
knowns, and  since  y=bt.     But  it  is  necessary  to  satisfy  also  the  inequalities 
20  — Gx— 3<>0,  20-4-fcr— 13*>0. 
In  adding  them,  x  disappears,  and  there  remains 

40— 16O>0  .•.  *<2|; 
then  the  values  which  wo  ought  to  give  to  t  are  1=0,  1,  2. 
With  the  value  t  =  0  we  should  have 

y=0,  z=20  — 6x,  u=204-G.r; 
and  we  see  that  we  can  mako  x=0,  1,  2,  3.     From  whence  result  for  the 
proposed  equations 


'x=  0 

'  x=   1 

rx=  2 

rx=  3 

v=  0           j 

r=20 
u=20 

•?/=  0 
z=14 
kU= 

y=  0 
z=   8 

w=32 

v=  ° 

^«=38. 

With  the  value  t=l  we  should  have 

y=5,  r  =  17— G.r,  t/=7  +  6x; 
and  the  only  admissible  values  of  x  are  x=0,  1,  2.     Thence  result  the  throe 
solutions 

x=  2 

y=  5 

z=  5 

a  =  19. 


z=ll 

«  =  13 


Finally,  with  the  value  (  =  2  we  should  have 

2/=10,  z=14— 6x,  u  =  —  G  +  Gx. 
The  only  admissible  values  of  .r  are  x=l,  2;  and  from  thence  result  the 
two  further  solutions 

( x=   1  (x=   2 

i/  =  10  I  i/  =  10 

z  =  8  ]  z  =  2 

u  =   0  (  u=   ti. 

In  all,  nine  solutions.     There  would  be  but  three  if  those  w  hided  in 

which  one  of  the  unknowns  is  zcm. 


1  \  \  M  1  I 

1°.  Two  countrymen  have  together  100  eggs.  The  one  says  to  the  other, 
If  I  count  ti iv  eggs  by  eight  1,  there  is  a  surplus  of  7.  The  second  answers, 
If  I  count  mine  by  tens,  1  find  the  same  surplus  of  T.  How  many  eggs  had 
each  .' 

Ans.  Number  of  eggs  of  the  first,  =63  or  23;  of  the  second,  =37  or  77. 

To  find  three  whole   numbers    BUCU   that,   if  we   multiply   tile    lir-t    by   3, 
tlie   second  by  .r),  and   the   third  by  7.  the  sum  of  the  products  shall  be 


QUADRATIC  EQUATIONS.  199 

and  such,  moreover,  that  if  the  first  be  multiplied  by  9,  the  second  by  25,  and 
the  third  by  49,  the  sum  of  the  products  shall  be  2920. 

Ans.  First  number,       =15  or  50. 
Second  number,  =82  or  40. 
Third  number,     =15  or  30. 
3°.  A  person  purchased  100  animals  at  100  dollars;  sheep  at  3J-  dollars  a 
piece  ;  calves  at  1 J  dollars  ;  and  pigs  at  J-  a  dollar.     How  many  animals  had  he 

of  each  kind  ? 

Ans.  Sheep,  5,  10,  15. 
Calves,  42,  21,  6. 
Pigs,      53,  66,  79. 

4°.  In  a  foundry  two  kinds  of  cannon  are  cast ;  each  cannon  of  the  first  sort 
weighs  1600  lbs.,  and  each  of  the  second  2500  lbs.  ;  and  yet  for  the  second 
there  are  used  100  lbs.  of  metal  less  than  for  the  first.  How  many  cannons 
are  there  of  each  kind  ? 

Ans.  Of  the  first,       11,36...;  of  the  second,  7,  23 ... . 
Or,  of  the  first,  ll  +  25£ ;    of  the  second,  7  +  16f. 
5°.  A  farmer  purchased  100  head  of  cattle  for  4000  francs,  to  wit:  oxen  at 
400  francs  apiece,  cows  at  200,  calves  at  80,  and  sheep  at  20.     How  many  had 
he  of  each? 

Ans.  In  excluding  the  solutions  which  contain  a  zero  the  problem  admits  of 
the  ten  following : 

Oxen,  1,  1,  1,  1,  1,  1,  1,  1,  4,  4. 
Cows,  1,  2,  3,  4,  5,  6,  7,  8,  1,  2. 
Calves,  24,  21,  18,  15,  12,  9,  6,  3,  5,  2. 
Sheep,  74,  76,  78,  80,  82,  84,  86,  88,  90,  92. 


QUADRATIC  EQUATIONS. 

178.  Quadratic  equations,  or  equations  of  the  second  degree,  are  divided 
into  two  classes. 

I.  Equations  which  involve  the  square  only  of  the  unknown  quantity. 
These  are  termed  incomplete  or  pure  quadratics.  Of  this  description  are  the 
equations 


rf 


ax*=b;  3a*+ 12=150-0?;   3— Y2+3a;3=2i  +  2:c2+~2T  ; 
they  are  sometimes  called  quadratic  equations  of  two  terms,  because,  by  trans- 
position and  reduction,  they  can  always  be  exhibited  under  the  general  form 

ax"=b. 
Thus  the  third  of  the  equations  given  above, 

x"      5       n  n_  7  259 

■3-  i2+3a?— M+^+'aT 

when  cleared  effractions,  becomes 

8x3— 10  +  72.i-=7+48.r2+259, 
or,  transposing  and  reducing, 

32.t2=276, 
which  is  of  the  form 

ax-=b. 


200  ALGEP. 

II.  Equations  which  involve  both  the  square  and  the  simple  power    f  the 

unknown  quantity.     These  are  termed  adjected  or  complete  quadratics.     Of 

this  description  are  the  equations 

x     3  2.r  273 

ax"-+bx=c;  z=-10.r=7;  —  --  +  -  =  8-j-x"-  +  —  ; 

they  are  sometimes  called  quadratic  equations  of  three  terms,  because,  by 
transposition  and  reduction,  they  can  always  be  exhibited  under  the  general 
form 

ax"-\-bx:=c. 
Thus,  the  third  of  the  equations  given  above, 

5.r2     x     3  2j       o     273 

~6~"~2+4  =  8  — "3  _ *°"+T2~' 
when  cleared  of  fractions,  becomes 

10a?—  G.r+9=9G  —  Sx— 12X-+273, 
or,  transposing  and  reducing, 

22x2+2x=360, 
which  is  of  the  form 

ax--\-bxz=c. 

SOLUTION   OF  PURE   QUADRATICS   CONTAINING  ONE   UNKNOWN   QUANTITT 

179.  The  solution  of  the  equation 

ax-  =  b 
presents  no  difficulty.     Dividing  each  member  by  a,  it  becomes 

b 


whence 


x~=-, 
a 


-±£ 


b 

If-  be  a  particular  number,  either  integral  or  fractional,  we  can  extract  its 

square  root,  either  exactly  or  approximately,  by  the  rules  of  arithmetic.     If 

b 

-  be  an  algebraic  expression,  we  must  apply  to  it  the  roles  established  for  the 

extraction  of  the  square  root  of  al  quantil 

It  is  to  be  remarked,  that  sinco  the  square  both  of  -\-m  and  — m  is  -j-m9. 


ova 


eo,  in  like  manner,  both  (+./-)    ™v\  (—    /-)    is  +  -.     Hence  the  ab 

equation  is  susceptiblo  of  two  solutions,  or  has  two  roots  :  that  i<.  there  are 
two  quantities  which,  when  substituted  for  x  in  the  origu  a]  equation,  will  ren- 
der the  two  members  identical  ;  these  are 

lb  (b 

.r=  -I-    /-  and  .r=  —    /-  ; 

for,  substitute  each  of  these  values  in  the  original  equat'on  ax',=b,  it  become* 


and 


aX\—Jj  ■**•  or  «X-=6i  i-  e.,  I>  = 


QUADHATIC  EQUATIONS,  oqi 

Henco  it  appears  that  in  pure  quadratics  the  two  values  of  the  unknown 
quantity  are  equal  with  contrary  signs.* 

EXAMI>LE   I. 

Find  the  values  of  a-  which  satisfy  the  equation 

4x-— 7  =  3x-2+9- 
Transposing  and  reducing,  .t-  =  16 

.-.  x  r=±  -/16 
=  ±4; 
henco  the  two  values  of  x  are  +  4  and  — 4,  and  either  of  these,  if  substituted 
for  x  in  the  original  equation,  will  render  the  two  members  identical. 

EXAMPLE   II. 

x*  5x"      7  299 

3~3+T2=24-'r2+"2T• 
Clearing  of  fractions,  8x-— 72+10.r2=     7— 24.t2-}-299 
Transposing  and  reducing,  42r2=378 

378 


i* 

"42 
=     9 

••.  X 

=  ±3, 

and  the  two 

values  of 

x  are 

+: 

5  and 

—3. 

EXAMPLE    III. 

3a2 

X2 

x  ■ 

=5 

5 

=3 

3 
Since  15  is  not  a  perfect  square,  we  can  only  approximate  to  the  two  va] 
of  a:.     We  find  the  approximate  values  to  be 

ar=1.290994,  or  —1.290994. 


EXAMPLE  IV. 

X 


^/ri-\-x"-—x 


-.m. 


Clearing  of  fractions,  x  =??i  V  rs+a? — mx, 

.-.  (m-\-l)x  =wiA//--+.r-. 
Squaring,  (m'i-\-2m-\-l)x"=m-(r--{-x-), 

.-.  (2m  +  l)x°  =  m"r2 

mr 


'  -/2m+l" 

*  One  might  suppose  that  in  extracting  the  square  root  of  both  members  of 
tion  as  afl=b,  the  double  sign  should  be  prefixed  to  x,    the    root  of  x",  also.    But  it  is  to 
be  observed,  that  it  is  the  value  of  -\-x  that  is  required.    Besides,  suppose  we  were  to  write 
-\zx=^^.V b :  combining  these  signs  in  all  possible  ways,  there  result  the  four  equations, 

■4-x=-f-v'£>  -f-.r= — t/1,  — 2-— -f--i/"'-  — .r= — ^b, 
the  last  two  of  which  may  be  deduced  from  the  first  two  by  changing  the  signs  of  the  two 
members  ;  the  equation  J-.r^-J-y'i  expresses  nothing  more,  therefore  than  the  equation 
ar=-J-l/&.     We  might  always  omit  -J^,  si  aplied  before  i/ 


202  ALGEBRA. 

i:xAJ:ri.i:  v. 


m-\-x-\-  \/-J',i  r-\-.i: 

=  11. 

m  +  r  —  V  ~>nx + x- 
Render  the  denominator  rational  by  multiplying  both  terms  c    the  fraction 
by  the  numerator,  the  equation  then  becomes 

(?«-f--r+  y/2mx-\-3?)* 

m2 
Extracting  the  root, 


ra-}-x-f-  ->/2mx-\-x-=±m^n. 
Transposing,  t/  2/nx-\-x-=  ^m  <J  n —  (m+ x). 

Squariag,  2mx-{-x-=m"n^2m  V«(»i+x)-|-(»i-}-;r)8. 

transposing  and  reducing, 

±2m.  i/  n(?n-\-x)=m2(l-}-n), 

m(l+    | 
.•.  m-f-x  s=- 


n) 
'n 


m(l-\-n) 
x  =  —m 


=  rLm 


2-y/n     ' 

(6)  ll(x2— 4)=5(x*+2).  Ans.  x=±3. 

^_j_7         x 7  7 

'7)  ta — = .  , ; — =^=0.  Ans.  x=4-9. 

x3 — 7x     x--f-7.r     x- — 73 


m-4-  -\/ mz —  x2     x 


(8) =-.  Ans.  x=±  y/2mn— »•. 


(9)    V^-^-  V>-+*:=P  Ang-  x=±     ;,«*<?,-.g)W(;,  +  0» 

•v/p+x-r-  -v/v — x        /x  

^10)  V* V?  =  0'  AnS-  *=±2Vj>q-f- 

180.  In  the  same  manner  wo  may  solve  all  equations  whatsoever,  of  any 
degree,  which  involve  only  one  power  of  the  unknown  quantity ;  that  is,  all 
equations  which  are  included  under  the  general  form 

ox  =  b, 
or  equations  of  two  terms. 
For,  dividing  each  member  of  the  equal  nos 

b 

x"=-. 
a 

Extracting  the  rik  root  on  both  sides,  , 

If  n  be  an  even  number,  then  the  radical  must  :ted  with  tho  double 

nign   ±,  for,  in  that  case,  both   (+M~)     M>d   (  —  "/-)     wiL  equally  pro 

duce  - 

a 


QUADRATIC  EQUATIONS.  203 

EXAMPLE  XI. 

5&—  57=2^+135 

3x6=192 

x"=64 

Here  +2  and  —2  are  two  of  the  roots  of  the  above  equatior 

EXAMPLE   XII. 


p       '       x  q 

px^Jx 

(p+x)Vl>+x=— —• 

2  3    p 

Or,  (p+x)L'=x2 .-. 


Squaring,  (^+x)3=a-3. - 


9 

/- 

f/1 


Extracting  the  cube  root,  p-\-x=x*  1L 


V 


EXAMPLE  XIII. 


T7  -p-i      r  :-i  a  /TV  — 

Ans.  x — '-  -•■>*-■» 


— Xq     =  -x 
q  S 

EXAMPLE    Xi  v  . 

G4?/5— 48?/4+127/2—  1  =-G  1. 
Extracting  the  cube  root,  we  have 


=  I  —  |p'— • 

w 


/5      V^ 


42/' 

EXAMPLE   XV. 
.T3  — 1/s  =  117 (1) 

x  -y  =    3 (2) 

Cubing  the  latter  equation, 

r3— 3x-?/+3x^/-— t/3=  o7, 
out  x3         —  T/3  =  117. 

.-.  by  subtraction,  3x2y—3xy'2        —  90, 

and  '  xy(x—y)  =  30; 

dividing  by  (2),  we  have  .-.  xy         =   10. 

Now  from  (2)  x3— 2xy+y*—     9, 

and  4.ry         =  40. 

.-.  by  addition,  x3+2xy+y'*=  49, 

and  k+J=±7, 

but  (2)  x— y  =     3. 

By  addition,  2x  =   10,  or  —  4, 

.-.  x  =     5,  or  —   2, 
and  by  subtraction  2y  =     4,  or  —10, 

.-.  y  =     2,  or  —  5. 


201  ALGEBRA. 

(16)  J=2a*+26,  Ana.x=±i/Ti, 

(17)  x- :  (18—*)*::  25  :  16.  Ans.  r=10  or  90. 

x       14 x  * 

(18)  HZ^: — T"::1  Ans.x=8or5G. 

75Qr-7)     48(z-4) 

^iJi  '  x_4  ~=     r_7    •  Ans.  x=19  or  J 

(20)    .  •_    /=  10,  iy— ^3—15.  Ans.  .r=±8,  y  = 

(•-'I)   (*— y)r=91,  (x— y)»=49.  .  x=±13,  t/=±6. 

(22,   (*-y£=24,  (*-y)£=6. 

z  =12,  or  i. 

(23)  x:y  =  13,  xy-=36.  Ans.  x=4,  y=3. 

(24)  £ry=  a/^+2/2+2-+2/,  a«+y«=:(:r+  .'/)*-  i 

A:  3.  T=6,  y= 

'-  ■/''  ?/.r24-?/-.r 

(25)  ;'r--.r+i/,  2_££_=4.  Al)S.  r=  2j  y==  , 

(26)  ia+y°=a,  xa-ya  =  b. 

(2;             ,r'  +  l0.r3— I0x-  +  5x— 1=32.  Ans.  r=3. 

(28)      -2*«+l=  Ans.  <=±V6- 

'   -  Vy=3,  V-r+  Vy=7.  Ans.  1=625,  y=16. 

(30)  .!-•-_ y'=369,  .r-  —  y*=9.  Ans.  z=   |  -,.  v=±4. 

(31)  .r5— t/3=56,  x—y=—.  Ans.  .r=4  or  —2,  y=2  or  — 1. 

!)  .r-.y  +  T/^HG,  xy?+y=14.  Ans.  .r=5  or  2-/|,  y=4  or  10. 

V*+  \fy=6,  x+y=72.  Ans.  x=64  or  8.  y=8  or  6 

34)         |       =20,  ar|+ yy=6.      Ans.  .r=±8  or  i  -/8,  .7=32  or  1 
(35)  .r4+2.r2?/-+y<=129G  — 4xy(x2+xy+ij:),  x—y= 

Ans.  .r=5  or  — 1,  ;/  =  l  or  —  5 
181.  We  have  seen  that  an  equation  of  the  form  ax"  =  b  1  roots,  or 

that  there  are  two  quantities  which,  when  substituted  for  x  in  the  original 
equation,  will  render  the  two  members  identical.     In  like  maimer,  we 
find  that  every  equation  which  involves  x  in  the  third  power  has  three  roots  ; 
an  equation  which  contains  x*  h  'oots ;  and  it  is  a  general  proposition 

in  the  theory  of  equations  that  an  equation  has  as  many  roots  as  it  has  di- 
mensions. 

:.  The  above  method  of  solving  the  equatii  zb  will  give  us  only 

one  of  the  n  roots  of  the  equation  if  n  be  an  odd  number,  and  two  roots  if  n  be 
an  even  number.     Such  a  solution  must,  therefore,  bo  c  1  imperfect, 

iliiiI  we  must  ha  arse  to  different  pp  to  obtain  the  remaining 

roots.     This,  however,  is  a  subject  which  we  in  one  for  the  present 

SOLUTION  OF  COMPLETE  QUADRATICS.  COITTAINII  '   \KNn\VN  iM\n 

183.  [n  order  to  Bolve  the  general  equal 

\-h.r=c, 

let  us  begin  by  dividing  both  members  bj     .  the  coefficient  of  .< ' :  the  o-jua 

u<m  then  becomes 

0        c 

-(--!■=-, 


QUADRATIC  EQUATIONS.  205 

or, 

x"-\-px=q, 

putting,  for  the  sake  of  simplicity, 

6         c 
a     1    a      l 
This  form  of  the  quadratic  equation  may  be  produced  by    multiplying  to- 
gether two  simple  equations.     Suppose 

x— a=0,  x— /3  =0; 
...  (a-a){x-(i)  =  0, 
which  is  satisfied  by  making  x  =  a,  or  x=3. 

Multiplying  the  two  factors  (x—a)  and  (i— /3),  the  equation  becomes 

x*—{a+(3)x+a8=0 (1) 

Substituting  first  a,  and  then  (3,  for  x,  this  may  be  written  either 

a2  —  (a+/J)a  +  a/3  =  0, 

or 

(3-  —  {a+(3)(3+af3=0, 

which  are  identical. 

Putting  in  equation  (1)  above,  p  in  place  of  —  (<z-f-/?),  and  —q  in  place  ot 
a.8,  it  assumes  the  form 

a-2-|-px — q=0. 

But 

_p2  =  a-+2a,3+/32 

—  47=         4a/?        < 

By  subtraction,  •  „  ,    . s — •-,  0  i  en     ~i       ««' 

.-.  a  — 8  =  V7;2  +  47- 

q+]3=— p. 

By  addition  and  subtraction,    a=  — 2P+3  v'i''2+45 

(3=-lp-iVp-+*q- 

As  a  and  /3  are  the  values  of  .r,  and  differ  only  in  the  sign  of  the  radical  part, 
both  may  bo  written  together  thus :  ' 

Hence  tho  following  rule  for  resolving  a  complete  or  adfected  quadratic 
equation. 

Reduce  the  given  equation  to  the  form  x3+px — q=0  by  clearing  of  frac- 
tions, transposing  all  the  terms  to  the  first  member,  and  dividing  throughout  by 
the  coefficient  of  the  square  of  the  unknown  quantity.  The  equation  being  thus 
prepared,  the  value  of  the  unknown  quantity  will  be  equal  to  ±  the  coefficient  of 
its  first  power  with  the  sign  changed,  ±?  the  square  root  of  the  square  of  this 
coefficient  — 4  times  the  knoivn  terms  of  the  equation. 

Tho  expression  x= — ^pzLlVp"-\--lq  may,  by  passing  the  |  under  the 
radical,  be  written  x=  —  l,p^z  V  {%?)*-{-&  which,  translated  into  a  rule,  is 
often  the  more  convenient  form. 

EXAMPLES. 

(1)  z2— yx+2=0. 
By  the  rule, 

11  ,       //11\3  11  ,        /121  11  ,  ,    /49      11  ,  7 

*=6-±K%)  -4x2=¥±n/-9--8=ir±Wir=-G  ±*« 
2 

.'.  x='3  or  -, 
according  as  we  use  the  upper  or  lower  sign. 


206  ALGEB 

(2)  3r— Is  =2  ;  changing  nil  the  si 
Xs— 3;r=  —  2,0      »— 3x+2=0. 

By  tho  rule, 

3 

X=5±J-/9— 2X4=s2  or  1. 

Hither  of  thcso  values  of  x  will  satisfy  the  given  equaticn.     First  substi- 
tuting 2,  wo  have 

3x2—4=2; 
and  substituting  1,  we  have 

3X1  —  1  =  2. 

(3)  x-+6z= 16. 
By  tho  form, 

x=—]p±  V{hp)-+q 

x=  —  3±  a/^+16=2  or  —  8. 

(4)  x-  — 10.r=— 21 

X—5±  V25— 21 
x=7  or  3. 

(5)  acx2-f  tax — adx — bd=0. 
Dividing  by  ac, 

}l>      <\        bd 
x2-f    -  —  -).r=  — . 
1   \a     c/        ac 

.-.by  the  rule, 


^-3W£¥+?-- 


.-.  x=-,  or . 

c  a 

(6)  a*+6x=27.  Ans.  x=3,  or  —9. 

(7)  x"-— 7.r+3|  =0  Ans.  x=6$,  or  1. 

10.r 

(8)  a?+— =19.  ins.  x=3,  or  — 6J. 

(9)  »=3+i2"  =10,  or  2. 

(10)  x2— Gx+8  =  80.  Ans.  x=12,  or  —6. 

(11)  x°—  10x+17  =  l  .  r=8,  or  2. 

(12)  x*— x— 40=170.  Ans.  x=15,  or  —11. 

(13)  3x2— 9x  — 4  =  80  \>    .     =7,  or  —4. 

(14)  7x2— 21.r+13  =  293.  Ans.  x=8,  or  -5. 

X*      4x  57 

(15)  -+7 19=15*.  Ans.  x=9,  or  — — . 

o           O  o 

2xq              x  9 

(1G)  —  +  3J- -+8.  Ans.  x=3,  or  — -. 

(17)  x-j- I-) =  13.  Ans.  x=  i,  or  —2. 

3G  —  u 

(18)  4u— =46.  Ans.  u=12,  or  —J. 

5—  i     9— 3p 

(19)  1G-- -Tr= -  +  3P.  Ans.  ..  =  •;.  . 

~  P 

16+3      1G—  G9 

<20>  HT+ *df=6*"  An..  y  =  5,  or -. 


QUADRATIC  EQUATIONS.  207 

□  _1_7  0  +  4D 

(21)  14+4  □  —77^7=3  D  + — g —  Ans.  D  =9,  or  28. 


Ans.  A =2,  or 


3 


(23)   ^-f -?=—;—.  Ans.  <=2,or|f. 


7A3      2A      11A  +  18 
(22)  —       3-=       33       • 

<  +  22       4       9< 

1  < 

<t>        0+1      13  t 

<24)  ^i+^T^T  Ans.  0=2,  or  -3. 

(^)  xT60=3-xb-  An3-  X=14'  "  -10' 

8?j  ^0 

(26)  — — -  —  6=—.  Ans.  v=\0,  or  —  §. 
'  tf+2             3u 

48           1C5  .               '   .       _ 

<27)  ^=7+To-5-  Ans.  »=53,  or  5. 

(28)  x3— 8x=14.  Ans.  x=9.4772+,  or  —1.4772+. 

(29)  3x3+x=7.  Ans.  x=1.3699+,  or  — 1.7032+. 

(30)  6x— 30  =  3x3.  Ans.  x=l  +  3  V  —  1- 

(31)  (x—  V  142.334) (.r+  -/ 142.334) =27.22x. 

Ans.  a:=13.61±  V  327.566. 

(32)  23  :  (140+x)  =  (240+x) :  1041.  Ans.  x=— 27.4  or  —352.6. 

,    V34 

(33)  (x+6):(3x+12)  =  (3x— 12):  (a:— 6).  Ans.  ar=±— £-. 

(34)  21::3— 1617z  +  20748=0.  Ans.  z=60.73,  or  16.27. 

(35)  3.5ga— 11.75g— 41.25=0.  Ans.  g=5.5,  or  —2.14. 

1+  V1008 

(36)  (3x+l)(4x— 2)  =  (13x+7)(5x— 3).  Ans.  x= — 

x  7 

(37)  -7-77;— ?=0.  Ans.  .r=14,  or  —10. 

'  x+60      ox— o 

w+4     7—w     4w4-7 

(38)  — ^—— -_=— ^—— 1.  Ans.  w=21,  or  5. 

v     '       3         w — 3  9  • 

15— p      12—3o  23»+60 

(39)  —~-Tp-^r=1]> *J—>  Ans.i;=3,  or  ftf. 

x+11     9  +  4.r 

(40)  — +       ,     =7.  Ans.  x=3,  or  —A. 

2?+9     4o— 3  3(7—16 

(41)  ^9t-+4i+3  =  3  +  ^8-  AnS'  ?  =  6'  °r       *' 

2.r  — 1      8— x2     x  ,. 

(42K = -4 — .  Ans.  x=2,  or  — V. 

****  3— x      2x— 2T2  3 

3               6          11  »               o        "R 

(43)  4- = — .  Ans.  x=3;  or  ff. 

y     >  6x— x-^x3+2x     5x  lI 

4x3+7x     5x— x3     4X3  a               o             9t 

(441  ! 1  _ — .  Ans.  x=3,  or  — |i. 

K     '        19      ^  3+x        9  "^ 

r»+2x3+8  .               .             ., 

(45)  —f. ^=x3+x+8.  Ans.  x=4,  or  —  y 

'    x;+x — 6 

ac  .              c±  y/<9— \ac 

(46)  cx-—b  =  (a  +  b)x*.  Ans.  s=      2(q+&) 

(47)  x3+tx+cx+at— ac+6c— a3=0.  Ans.  x=a  — 6,  or  —  a—  c. 


ops  ALGEBRA. 

(48)  2(o-c)y/2-r-a»=(6-c)«+< 


Ans.  ?/  = ~ • 

184.  If  t=a  in  the  general  form  (.r— a){x— i)=0,  it  assumes  the  partic 
ular  form  (x—a)-=x"—2ax+a-=0. 

If  tho  two  values  of  .r  be  -\-a  and  —a,  the  form  (.r— fl)(.r+a)=.r-  —  a-  =  l). 

185.  Recollecting  that  the  value  of  the  unknown  quantity  is  called  the  root  of 
the  equation,  it  is  seen  that  every  equation  of  the  second  degree  has  two  roots, 
and,  by  the  general  form  (I),  x':  —  {a-{-b)x-\-ab=0,  that  their  sum  is  equal  to 
the  coefficient  of  the  second  term  with  the  contrary  sign,  and  that  their  prod- 
uct is  equal  to  the  absolute  term  or  known  quantity,  when  transposed  to  th< 
first  member.  Thus,  in  Example  4,  above,  the  sum  of  the  two  roots  3  and 
_9  js  _(j?  and  tho  product  —27.  The  same  may  be  seen  in  other  exam- 
ples. 

The  general  form  ax"-\-bx=c  is  capable  of  producing  all  the  particular 
forms  by  the  supposition  of  particular  values  for  the  coefficients.  Thus,  if 
6=0,  it  assumes  the  form  of  pure  equations.     If  c  =  0,  it  may  be  written 

x(ax-[-b)=0, 

b 
which  we  perceive  may  be  verified  by  making  x=0,  or  ax-{-b=0  .-. x=—  -. 

b 
The  roots  are,  therefore,  in  this  case,  0  and  — -.     Whenever  an  equation  is 

divisible  throughout  by  the  unknown  quantity,  one  of  its  roots  is  zero. 

When  wo  know  that  the  two  roots  of  the  equation  of  the  second  degree  are 
real,  the  above  relations  make  known  at  once  the  nature  of  these  roots ;  for 
example,  admitting  that  those  of  the  equation  x-  — 2x— 7  =  0  are  real,  we 
conclude  immediately  that  they  are  of  different  signs,  because  their  product 
is  equal  to  the  absolute  term  —7,  and,  moreover,  that  the  greater  is  positive 
because  their  sum  is  +2,  the  coefficient  of  x  taken  with  the  contrary  sign. 

18G.  Another  mode  of  solution  may  be  derived  as  follows  : 

If  wo  can,  by  any  transformation,  render  the  first  member  of  the,  equation 
x?_j_p;r=<7  the  perfect  square  of  a  binomial,  a  simple  extraction  of  the  square 
root  will  reduce  tho  equation  in  question  to  a  simple  equation. 

But  (.r-H  j;)3  is  x--\-jrx-{- ' 

In  order,  therofore,  that  tho  fust  member  may  be  transformed  to  a  perfect 
square,  wo  must  add  to  it  the  square  of  './> ;  that  is,  the  square  of  half  the  co- 
efficient of  the  second  term,  or  simph  \  ;  it  thus  becomes 

x"-+]u+lr 

which  is  the  square  of  x-f  _,.      Bui   sine,'  \v  I        to  the  left-hand 

member  of  the  equation,  in  order  thai  tin-  equality  between  the  two  members 
may  not  be  '1  troyed  wo  must  add  tin-  same  quantity  to  the  right-hand  mem- 
ber also  ;  the  equation  thus  transformed  will  be 


QUADRATIC  EQUATIONS.  20'J 

or 

p        Ip2 

Extracting  the  root,  x-\-  -  =  ±  -y  ^-+  ?• 


Transposing,  x=—-±yl-—-\-q 


*=- fWf 


— JP  ±  \Ap2  +  4g 
-  2 

the  same  form  for  the  value  of  x  as  we  obtained  by  the  first  method. 

If-  \~f~ 

We  affix  the  sign  ±  to  ■\f-r-\-Qt  because  the  square  both  of  -\-yJ-r-\-q- 

and  also  of  — \/~r  +  ?>  *s  +(x"^^)'  an(^  eveiy  quadratic  equation  must 

therefore,  have  two  roots. 

From  what  has  just  been  said,  we  deduce  the  following  general 

RULE  FOR  THE  SOLUTION  OF  A  COMPLETE  QUADRATIC   EQUATION. 

1.  Transpose  all  the  known  quantities,  when  necessary,  to  one  side  of  the 
equation,  arrange  all  the  terms  involving  the  unknown  quantity  on  the  other 
tide,  and  reduce  the  equation  to  the  form  ax2-f-bx=c. 

2.  Divide  each  side  of  the  equation  by  the  coefficient  ofx". 

3.  Add  to  each  side  of  the  equation  the  square  of  half  the  coefficient  of  the 
simple  power  ofx. 

That  member  of  the  equation  which  involves  the  unknown  quantity  will 
thus  bo  rendered  a  perfect  square,  and,  extracting  the  root  on  both  sides,  the 
equation  will  be  reduced  to  one  of  the  first  degree,  which  may  be  solved  in 
the  usual  manner. 

EXAMPLE  I. 

12x— 210=205— 3.r2+5. 
Transposing  and  reducing, 

3.r2+12r=420. 
Dividing  by  the  coefficient  of  a;2, 

2;2+4a;=140. 
Completing  the  square  by  adding  to  each  side  the  square  of  half  the  coefficient 
of  the  second  term 

z2+4:r+4  =  140  +  4, 
o 

(,r+2)2=144. 

Extracting  the  root,  x-f-2=±  -\/l44 

'     =±12 
.-.  x=—  2  ±12. 
Hence 

<x=— 2-f  12  =  10 
(x=—  2  — 12  =  — 14. 
Either  of  these  two  numbers,  when  substituted  for  .r  in  the  original  equation, 
will  render  the  two  members  identical. 

O 


210  ALGEBRA. 

EXAMPLK    II. 

2a:2+34  =  20x+2. 
Transposing  and  reducing, 

2a:2— 20x=— 32. 
Dividing  by  2,  a:2  —  10z=  — 16. 

Completing  the  square, 

a?— 10x+  25=25— 16, 
or  (x— 5)2=9. 

Extracting  the  root,  x — 5=  i  V9- 

r=5±3. 


Hence 


<a:=5+3=! 
(x=5  — 3=', 


EXAMPLE    III. 

3a:2  — 2a  =  65. 

2       65 
Dividing  by  3,  x-—-x=z—. 

Completing  the  square, 

3  +  \3/  ~~  3  +  \3/  ' 


or 


(*-$=- 


96 


Hence 


-±¥ 

1      14 
XBB3±T 

1+14 
x=-^-=5 

1—14  1 

X=-3~  =  -43- 

EXAMPLE  IV. 

rS+x— 2  =  0. 
Transposing,  z2+x        =  2. 

The  coefficient  of  x  in  this  case  is  1 ;  .-.  in  order  to  complete  the  square.  w*» 

/iy      1 

must  add  to  each  side  I -I  ,  or  -. 


•'•  *M-*+J= 

=  24— 
~t4 

W- 

9 
=  4 

1 

X+2  = 

■±l 

.'.  x=l,  and  ars 

o 

QUADRATIC  EQUATIONS. 


211 


EXAMPLE  V. 

6x— 30=3x2. 
Transposing,  — 3x'2+6x=30. 

Changing  the  sign  on  both  sides,  - 

3x3— Gx=  —  30. 
Dividing  by  3,  x2— 2x=  — 10. 

Completing  the  square,    x2 — 2x-|-l  =  l — 10, 
or 

(x— 1)"=  —  9. 


Hence 


x— 1  =  ±  V  —  9. 


(x=l  +  V— 9 
?x=l— •/— 9' 
In  the  above  example,  the  values  of  x  contain  imaginary  quantities,  and  th« 
roots  of  the  equation  are,  therefore,  said  to  be  impossible. 


EXAMPLE    VI. 

5         1        3      ,      2  273 

-^--x+-  =  8-3-,-^+— . 

Clearing  of  fractions, 

10x2— 6x4-9=96— 8x— 12x2-f  273. 
Transposing  and  reducing, 

22x2+2x=360. 
Dividing  both  members  by  22, 

2        360 
22'T:="22 

/1\2 
Adding  I  — )    to  both  members, 

i2 


*2+™*=l^-- 


a:2+22:r+  \22/  ==22_+\22/ 


Extracting  the  root, 


Hence 


ar+22=±V22"+W) 

/7921 
=      V(22)» 


~±22' 


1       89 
x=— — 4- — =4 

22^22 

1       89_ 
X  22~"22~~ 


45 
"11* 


Transposing, 


ax2- 


EXAMPLE  VII. 

ac 


'a  +  b~ 


:CX —  bx*. 


(a-\-b)x~ — cx= 


ac 


a+b' 


n2  ALGEBRA. 

c  ac 

Dividing  by  a+b,        x°—-^+b  '  X=z{a  +  bf 

Completing  the  square, 

c  c»      ,  cc         c2 

xV~^p  '  X+4(a+t)a-(a+6)^4(a  +  6^' 

or 

r  c         )  9       c"4-4ac 

lX"2(a  +  6)S   ~4(a+6)»" 

Extracting  the  root, 


a: — 


c  •v/c-4-4gc 

2(a  +  6)  =  ±    2(«  +  6)' 


•••  x=      2(a  +  6)      ' 
The  two  values  of  x  here  are 


C_L.  Vc2+4ac      _c— jv/c^4-4ac 
*=      2(a+6)      'X-      2(a+6) 

EXAMPLE  VIII, 

m2x2 
a2+&2-2&x+x2=-^-. 

Transposing,       (n9— m^x2— 2*m2x=  _n2(a2-j-i2). 
Dividing  by  the  coefficient  of  x2, 

2bn*x  9    <*"+&' 

x°"—n2— m2— — W  '»»— m9" 

Completing  the  square, 

2fcn2x       /    bn2    \2_  /    6n9    \2 _7T-(a2+fr3) 
**— ^^Z^>+ Wll^2/  "~\n9— »»V         »9— to9 
or 

— \  m-{a:4-L-)—n:a-  [ 

Extracting  the  root, 

I__^L_=±-^— ■v'>n9la*+&9)-n9a9 


n2— m* 
x 


n2 —  m-  (  * 


The  two  values  of  x  are 


x=^^  |  bn+  /m9(a9+&»)-»"a«  \ 

n       S  I 

x  =  - „  J  l,n—%/m:(a--!rb-)  —  ri2ai  t 

n-  —  mr  t  ' 

.g\  x24-4x=21.  Ans.  x=3,  x= — 7. 

(10)  x2— 9z+4j=0.  Ans.  x=8j,  x=7y 

(11)  622x— 15x2=6384.  A.M.  x=22|»  *=18  . 


(12)  a-r9— 7x4-34=0.  Ans.  x: 


7+-v/_ln  7—  V— 1039 


z 


16  '  16 


aUADltATIC  EQUATIONS.  213 

—  1+V^33  _l_s/"l33 

(13)  3a.a+x=ll.  Ans.  x= - ,  x= - 

x  4x2 

(14)  -_ 4_ i»+23?— —=45  — 3j*+4:r. 

o  O 

6x2— 40      3r— 10  23 

(15)  3x — — — - — — =2.  Ans.  :r=— ,  x=4. 

v     '  2x — 1        9 — 2x  2 

90        90  27  5 

(16)  ——-—-r———7=0.  Ans.  x=4,  z=  —  -. 
v     '    x      x+1     x+2  3 

3a2a:     6a2+a&~2Z>2     Z>2.r  2a— b  3a+26 

(17)  ate2 + = - .      Ans.  x= ,  x= -7 . 

v     '  '     c  c2  c  ac  oc 

■\frnn  -\/mn 

(18)  m.f2 — 2mx-Jn=?ix'i — mn.         Ans.  £=— -. — ; — j-,  x=—: j-. 

v     '  y m-\-  yn  ym —  yn 

(19)  4o%2+4a2c2ar+4aii2x— 9c^2x2+(ac2+ic?2)2=:0. 

ac*+bd*  ac^+bd1 

Ans.  x= — - — ,      ■    ,  ,  x=- 


2a  +  3rtVc'  2a—Zdy'c 

5a  +  10a&2         (by'a  +  b      (l  +  26sW^c\        ci 

<20)  9T2^3^^-(^-rL-+        3-a2        )*+^(a+»>c=0- 

_(3— a»)Va+6      _3b*cdy'c 
AnS'  X_     afc(l+262)     ' X=      6a      ' 

187.  The  above  rule  will  enable  us  to  solve,  not  only  quadratic  equations, 
but  all  equations  which  can  be  reduced  to  the  form 

x-a-\-pxn=q; 
that  is,  all  equations  which  contain  only  two  powei's  of  the  unknown  quantify, 
and  in  which  one  of  these  powers  is  double  of  the  other. 

For  if,  in  the  above  equation,  we  assume  y=x",  then  y^^zx*1.  and  it  be- 
comes 

y*+py  =  q. 
Solving  this  according  to  the  rule, 


— pAi  V^'2+4? 

y= 5 • 


Putting  for  y  its  value, 


xn—— ?  ^  ^  +  4? 


2 
Extracting  the  nth  root  on  both  sides, 


V  o 


EXAMPLE  I. 

a-4— 25x2=— 144. 
Assume  x'i=y,  the  above  becomes 

2/2_25?/=— 144. 
Whence  2/=16,  2/=9. 

But  since  x"=y  .••  x=  ±  Vy  » 

.-.  a-=  ±  y/16,  x=  ±  V9- 
Thus  the  four  values  of  x  are  +4,  — 4,  +3,  — 3. 


214  ALGEBRA. 

EXAMPLE    II. 

x*—7x*=8. 
Assume  x-=y,  y"1 — 7y=.8. 

Whence  y  =  8,    y=  —  1 

And  since  x2=y  •'•  *=  i  V  y- 

Whence  the  four  roots  of  the  equation  are  i  -/8,  i  •/ —  1,  t.f  e  last  two 
of  which  are  impossible  roots. 

EXAMPLE  III. 

Let  x6— 2r»=48. 

Assume  z-^zzzy,  the  above  becomes 

i/2— 2jr=48. 
Whence  2/=8,  or  — 6. 

But  since  xs=y  .•.  x=  tyy. 

Hence  two  of  the  roots  of  tho  above  equation  are  -4-  1/8  and  —  1/6;  tti»- 
remaining  four  roots  can  not  be  determined  by  this  process. 

EXAMPLE  IV. 

Let  2x— 7  •v/i-=99, 

or  2x— 7x*=99. 

This  equation  manifestly  belongs  to  this  class,  for  the  exponent  of  .r  in  the 
first  term  is  1,  and  in  the  second  term  half  as  great,  or  .',. 

In  this  case  assume  ■y/x=y,  the  equation  becomes 

2y2—7y=99. 

Whence  2/=9,     y  =  —  tt. 

But  since  -\/x—y  .-.  x=y2 

121 

•  '.  2'z=bl,     Xzzz — - — • 

To  account  for  the  two  values  of  x  in  this  equation,  it  must  be  observed  that 
one  belongs  to  -\-  -\/x,  the  other  to  —  -y/t' 

This  will  appear  clearly  in  the  following  example. 

example  v. 

ax=b-\-  -v/cx (1) 

Solving  this  equation  in  the  same  manner  as  the  preceding,  we  shall  find 
2ab-\-c-\-  ■v/4atc-|-c-  2<7?>  +  c  —  <J\abc-\-c'i 

T=  2^  ,X=  2aT~  ' 

If  wo  substituto  these  two  values  of  x  in  the  original  equation,  we  shall  find 
that  the  first  only  will  verify  it ;  the  second  belongs  to  the  equation 

ax=b — -\/cx (2) 

These  two  equations,  multiplied  together,  produce  the  complete  quadratic 
equation 

<»V-  (2«/>  +  r).r+6«=0, 
whoso  roots  are  tho  two  valnes  of  x  given  above. 

Tho  explication  of  this  matter  Is,  that  \/x  is  always  supposed  to  have  the 
double  sign  i,  and  therefore  the  general  form  expressed  !>y  equation  (1)  in- 
volves covertly  that  expressed  by  equation  (2).     It  i  :iy,  therefore,  it- 


aUADRATIC  EQUATIONS.  215 

examples  of  this  kind,  to  tiy  the  answers  obtained,  by  substituting  diem,  m 
order  to  see  which  belongs  to  the  given  form. 

188.  Many  other  equations  of  degrees  higher  than  the  second  may  be  solved 
by  completing  the  square ;  although,  it  must  be  remarked,  wo  can  seldom  ob- 
tain all  the  roots  in  this  manner.  The  transformations  to  which  wo  subject 
equations  of  this  nature,  in  order  that  tho  rule  may  become  applicable,  depend 
upon  various  algebraic  artifices,  for  which  no  general  rule  can  be  given.  The 
following  examples  will  serve  to  give  tho  student  some  idea  of  the  course  he 
must  pursue ;  a  little  practice  will  soon  render  him  dextrous  in  the  employ 
ment  of  such  devices. 

EXAMPLE  VI. 


Let  Vz+12+\/i-+12  =  6     * 

Assume  x-j- 12=?/,  the  equation  then  becomes 

y*+y*=6,  . 

which  evidently  belongs  to  the  same  class  as  the  previous  examples ;  completing 

the  square,  we  shall  have 

i 
2/T=2,  or  —3. 

Raising  both  sides  of  the  equation  to  the  power  of  4, 

y  =  16,  or  81 

••.  x,  or  y  — 12=  4,  or  69. 

EXAMPLE  VII. 


Let  2a;24--v/2.r2+l  =  ll. 

Add  1  to  each  member  of  tho  equation,  it  becomes 

W+l-^  V2x3+l=12. 
Assume  2.t3+l=2/>  thon 

y+y*=\2. 

Completing  the  square,  and  solving,  we  find 

i 


2/2,  or  -v/2i-2+1=3'  and  — 4 
2a:24-l=9>  and  16 

,  15 
r2=4,  and  — . 


/l5  /l5 

Hence  x  =+2,  —2,  +\/~>  — VIP 


It  may  be  remarked,  that  it  is  in  general  unnecessary  to  substitute  y,  which 
has  been  done  in  the  above  examples  for  the  sake  of  perspicuity  alone. 


EXAMPLE  VIII. 


/        8\2  8 

Let  (x+-)  + x=42--. 

Transposing  V+x)  +  r+i)=42, 

g 
Considering  x+~.  as  one  quantity,  and  completing  the  square, 

/       8\3      /       8\      1      169 


216  ALGEBRA 

8  1,3 

•••x+i=-2±y 

=  G,  and  — " 

Hence  we  have  the  two  equations 

x" — Gxr= — 8 

x2+7x=  —  8. 

Solving  the  first  in  the  usual  manner,  we  find 

x=4,  and  2, 

and  by  the  second,  we  have 

— 7+-/17        ,  —  7  —  Vl7 
x= ,  and - , 

which  are  the  four  roots  of  the  proposed  equation.  If  we  had  r^'uc.ed  Una 
equation  by  performing  the  operations  indicated,  instead  of  employing  the 
above  artifice,  it  would  have  become 

z'+x3— 26x2+ 8x+64 =0, 
a  complete  equation  of  the  fourth  degree. 

The  roots  of  equations  of  the  fourth  degree,  reducible  to  the  second  as  abore, 

present  themselves  ordinarily  under  the  form  "\JaJc  Vb,  and  frequently  of 
ford  an  application  of  the  process  exhibited  at  (Art.  104). 

(9)  x<+4x2=12.  Ans.  ar=±-/2,  or  ±  V~6. 

(10)  x°— 8x3— 513=0.  Ans.  r==3,  or  —  ^"19. 

(11)  x*— 13x2+36=0.  Ans.  t=±2,  .r=±3. 

(12)  (x*-2)2=-(x2-fl2).  Ans.  x=±2,x=±-}. 

(13)  (a*— l)(x2— 2)  +  (x*— 3)(x2— 4)=x*+5.       Ans   s=±l,  x=±3. 

/m±  ■/'»-+ 4/AL 

(14)  x-°— 7nxn=p.  Ans.  x={  0    ^  J  ]". 

jte+2     4-Vx 

•  15)  — =s =— .  Ans.  .t=4  * 

4-f-  V^  V^ 

(16)  V^#==£=Vf.  An8    r=/-ft=fci/4*+4a»+ty 
a+  a/i          V-r  \  2(a+l)  /  * 

(17)  Vx3— 2 -v/x— -r=0.  Ans.  x=4. 

(18)  a/^+ -/^=6  v^  Ans.  x=2. 

(19)5=22^+^.  Ans.  x=49. 

3a/^ 
i o 

5  1 

(20) —  — — =0.  Ans.  x=25  or  49. 

v     '      x — 5         20 

(21)  x7+x-J=756.  Ans.  x=243,  or  (— 28) 

1  3 

(22)  Xs— x2=56.  Ans.  x=4,  or  (—  7) '. 

i  i 
*  In  this  and  sorao  of  the  following  examples  another  value,  *=— .  is  also  found,  bat  if 

•rill  not  satisfy  thu  equation,  and  is,  therefore,  to  b<  p.  -14.] 


QUADRATIC  EQUATIONS.  $217 

6 
5  5  /  —  Q7\ 7 

(23)  3x^+^=3104.  Ans.  x=G4,  or  I — -L\   . 


(24)  aJ+lJ=c.  Ans.  *=(*  ^^-by 

8  3 

4      /ir3"  /       74\  *~ 

(25)  3r3— —  =—592.  Ans.  x=8,  or  (_  — ) 

2  \       5  / 

n  2 

(26)  xn— 2ax*  =  b.  Ans.  x=(a±  y/a*+b)a 
a:i-f-4i  \/7c_A  [rr\/x-\-x)               2x* 
5\/x — x          3 — \/x        (5\/x — x)(3 — \/x) 

x  x  b 

■\Zx-\-\/'a — x     \/x — \/a — x     \/x 


r — a;  3 — \/x        (5\/x — x)(3 — \/x) 

m        * ,+  _  * =A.  Ans.  .J&IZ+* 


2 


,„„.  x+\/x2— 9      ,  %  8-4-1/— 11 

(29)     ^^  =(x—  2)2.  Ans.  *=5,  or  3,  or  ■ 


a;— -v/x2— 9 


j 


(30)  a;+5=-/a;+5+6.  Ans.  *==4. 


(31)  a;+16— 7-/-c+16=10— 4-/a;+16.  Ans.  x=9. 


(32)  i/x+12+i^o;+12=6.  Ans.  a;=4. 

(33)  x°—  2a:-f6-l/a:2— 2x-f  5=11 .  Ans.  as=l,  or  lj-2i/l5. 

(34)  2r2+3a:— 5-v/2^+3a;+9-f-3=0.  Ans.  a:=3,  or . 

(35)  [(a:— 2)2— a;]2— (a;— 2)2=88— (a;— 2).  Ans.  x=6,  oi  —1,  or  5±3V— 3 

(36)  (ar+6)2-f2x*(a;+6)=138+a;*.  Ans.  x=4 

o 

(37)  x—  l=2+-^r.  Ans.  x=4. 


f33)  art— 2a^+o;=132.  Ans.  a;=4,  or  —3,  or  ^^ — — 

4  1  9-J--/48I 


(39)  9x-f--/l6a:3-|-36.c!=15a3 — 4  Ans.  .r=-,  or ,  or  x= 


3'  3  50 


t*n\         12+8.rJ                                                                             .              „        — 34-V— 7 
(40)  x= — ^ .  Ans.  a;=9,  or ±-*- . 


X — 5 


,     .   49a;2  ,  48                  ,  6  8        —3-1-1/93 

41    — — U-_49=9+-.  Ans.  jc=2,  or ,  or ±V  -. 

y     '      4      rxi                ~x  7                 7 

(42)  ■— 1 — ^—I7x=8.  Ans.  x=4-2,  or  —8,  or  — -. 

2        4  -1-  '             '           2 

(«)  (-DM*-^!-  '   *■— i-v^ 

(44)  ar<— (24c+4n2)xs-j-J2t.2=o.  Ans.  a;=-J-\/^+2«'i2av'^c+c3. 

(45)  a;2-a;+5v/lai=5aT+6=52±-.  Ans.  JC=5=fc^1329>  ^  5C=3j  ^  _J 


(46)     ^ =-.  Ans.  ar=5(±l/-7-3). 


x— i/x'2— ai     « 


8' 


Note. — In  some  of  the  above  examples  we  have  given  answers  which  will  not  satisfy 
the  equation  unless  the  double  sign  be  understood  before  the  radical.  In  some  cases  this 
Bign  is  understood,  in  others  not;  but  whether  it  is  or  not  will  always  be  known  from  the 
probloia  from  which  the  equation  is  derived. 


218  ALGEBRA. 

ON  THE  SOLUTION  OF  QUADRATIC  EQUATIONS  CONTAINING  TWO 

KNOWN  aUANTITE 
189.  An  equation  containing  two  unknown  quantities  is  said  to  be  of  the 
second  degree  when  it  involves  terms  in  which  the  sum  of  the  exponents  of  the 
unknown  quantities  is  equal  to  2.  but  never  exceeds  2.     Thus, 

3x-_ 4.r+i/2— xy— 5^+0=0,  7xy—ix-\-y=0, 
are  equations  of  the  second  degree. 

It  follows  from  this  that  eveiy  equation  of  the  second  degree  containing 
two  unknown  quantities  is  of  the  form 

ay'2-{-bxy-\-cx--lrdy-{-ex-\-f=0, 
where  a,  b,  c, represent  known  quantities,  either  numerical  or  alge- 
braical; i.  c.,  the  equation  contains  the  second  power  of  each  of  the  unknown 
quantities,  the  first  power  of  each,  and  the  product  of  the  two.  Not  that 
every  equation  of  the  second  degree  contains  all  these,  but  when  any  one  of 
them  is  wanting  the  coefficient  of  that  term,  in  the  general  form,  is  said  to  be 
zero.  *. 

Let  it  bo  required  to  determine  the  values  of  x  and  y,  which  satisfy  the 
equations. 

arf+bxy+cx'+dy+ex+f  =0 (1)  ) 

ary*+b'xy+c?x*+d'y+e'x+f'==0 (2)  S 

Arranging  these  two  equations  according  to  the  powers  of  y,  they  become 

atf+{b  x+d)y+(cx"-+ex+f)=0 

ay-+(b'x+d')y  +  (c'x°-+c'x+f')  =  0 

Put  bx+d=h;    cx-+ex+f  —k 

b'x+d'=h';  c'x"-+e'x+f'=kf. 

.:ay"+hy  +  k=0 (3) 

ay+h'y+k'  =  0 (4) 

Multiply  (3)  and  (4)  by  a'  and  a  respectively,  and  also  by  fc  and  k ;  then 

aa'y"--\-a'hy+a'k=0 (5) 

aa'y"-\-ah'y+ak'=0 (6) 

aky+hk'y+kk'=0 (?) 

a'ky--\-h'ky+kk'  =  0 (8) 

Subtracting  (6)  from  (5),  and  also  (7)  from  (8),  we  have 

{a'h—ah')y-\-a'k—ak'  =  0 (9) 

(a'k—ak')y-\-h'k—hk'=0 (10) 

Multiplying  (9)  by  h'k—hk',  and  (10)  by  a'k—ak',  wo  have 
(a'h—ah'){h'k-hk')y+(a'k—ak'){h'k-hk')  =  0  .  .  (11) 
{a'k-akjy+{a'k-ak')(h'k-hk')  =  0  .  .  (12) 

A  (a'h—ah'){h'k-hk')  =  {a'k-aky (13) 

Substituting  the  values  of//,  h',  k,  lJ  in  equation  (13),  we  have 

=  I  (o-c— a<0i3+(o'«— flO*-H,l/*— «/'  1 2 
Hence,  by  multiplying  and  expanding,  tho  final  equation  in  x  is  of  the  fourth 
degree,  which  will,  in  general,  bo  tho  degreo  of  the  equation  obtained  by 
eliminating  between  the  two  equations  of  the  second  degreo;  but  the  general 
form  includes  a  variety  of  equations,  according  to  the  values  of  the  coefficient 
,r.  6,  c,  &c.;  when  d,  e,f,  d',  • '../"  are  each  =0,  the  solution  may  be  obtain- 
ed by  quadratics,  the  resulting  equation  in  x  beii 

{(a'6_oo')z+a'd— ad'\  .  {(fc'c— b*)x— {dd— cd')}s=(afc— tuff* 


QUADRATIC  EQUATIONS. 


219 


Although  tho  principles  already  established  will  not  enable  us  to  solve  equa- 
tions of  this  description  generally,  yet  thore  are  many  particular  cases  in 
which  they  may  bo  reduced  either  to  pure  or  adfected  quadratics,  and  the 
roots  determined  in  the  ordinary  manner. 


EXAMPLE   I. 

* 


Required  the  values  of  x  and  y,  which  satisfy  tho  equations, 

S  *+y=p 

(        xy  =  cf 

3?-\-2xy-\-y*ss:p* 

4xy =4<72 .  .  .  , 


Squaring  (1), 
Multiply  (2)  by  4, 


Subtract  (4)  from  (3),    x2— 2xy-{-y2—i)2— 4q°, 


or 

Extract  the  root, 
But  by  (1), 
Add  (1)  to  (5), 
Subtract  (5)  from  (1), 


{x—yf=p2—4q2. 
x— y=±  Vp3- 

x+y=p- 


■  4<f 


2x=Jp±  Vp"  —  4-q"~ 


Hence  the  corresponding  values  of  x  and  y  will  bo 


x=- 


P+  /?3-4<?2 


y— 


P —  Vp~ — 4<?2 


x= 


P—Vp2—4q"' 


•  and 


T- 


p-\-  Vp'2— 4?2 


(1)? 
(2)S 

(3) 
(4) 


(5) 


EXAMPLE  II. 


Square  (1), 
But  by  (2), 
Subtracting, 


c  x  -\-y  :=a  . 

\  X2+7/2=&2 

x3-f  2xj/  +  9/2=a2. 
x"  -\-y"'=b2. 


2xy        =a2— Z>2 
Subtract  (3)  from  (2),    x2— 2xy+y"=2b"~— a", 
or  (x— y)-=2¥— a2.     * 

Extracting  the  root,  x— 1/=±  -\/2b2 — a-. 

But  by  (1),  x+y=a, 


adding  and  subtracting 


2x=a±  \/2i2— a- 


2y=a^f  ■^2b2—ai. 
Hence  the  corresponding  values  of  x  and  y  will  be 

a+  V2b2— a2') 
x= - ^  x 

and 


a—  V262— a'- 


y=- 


a—  J2b2—a2 


y=- 


«+  j2b2— a2 


1 
J 


(3) 


EXAMPLE   III. 


Cube  (1), 
But  by  (2), 
Subtracting, 
or 


<x  +  y  =m.  .  .  . 
( x3-\-y3=?i3    •  •  • 
x3+  3x2y+ 3xt/2+ y3 = ?n3. 

X3  -4-?/3=tt3. 

3x2y-\-         3xy2—tri3 — n3, 
3xy{x-\-y)=?n3 — n3. 


an 

(2)V 


220 


ALGEBRA 


Substitute  for  (x-\-y)  its  value  derived  from  (]), 

3xy  .  m  =  /ir  —  n3 


xy. 


4xy-. 


3m 

4(w3— n  ) 


3?7i 


Squaring  (1), 
But  by  (.3) 


Subtracting, 


or 


But  by  (1), 


x2-\-2xy-\-y2=mr-. 


•1/7/         = 


4(m3 — n3) 


3  m 


x2 — 2xy + y2 = m2 


4(m3— n3) 


3m 


(x-y)°- 


An3 — mz 
3m 


-.x—y    =±yj~ 
x+y    —m 


■711° 


3m 


•.  2x=m±J— 


3m 


—  ni- 


ne' 
m 


Hence  the  two  corresponding  values  oTx  and  y  aro 

_m        Un3  —  mA  m        hn3- 

i — ; f  and  , f 

m        Un3—m3  m  An3—m3 

y-2-\J-T2m-)  ^2  +  V-T^-J 

EXAMPLE  IV. 

$**+**y*+y*=a ( 

1  3    3 

C.r3^_xL.2/2_|_3/3=7J ^ 

square  (1),  tf+xtyS+tf+'Hx11 .  X*y*+2x*y5+2y1  .  A"=a» 
But  by  (2),  a^+gy-fy8  -_/,. 

Subtracting, 


1 

2W 


,.*,,* 


nr 


2x-  .  xV+2a:V+2yTxV=«'— *, 

5     3       3  3     3  3 

ryV+iy+yt)=(i,-i 


But  by  (1), 
And  by  (3), 

Adding, 

or 


.•.  2.r'  y*  .  a 
2         •'<  3 


2a 


(3) 


2a 


3  3     :t 


<l'  — 6 

'~~2u~' 


j       a       3a9  +  6 

(xT  +  1/  4)5  =  — 


2a 


•'•'+.'/' 


f=±v^ 


(«> 


QUADRATIC  EQUATIONS. 


221 


Again,  from  (1), 
And  from  (3), 

Subtracting, 

01 


But  by  (4), 


3  3     3  3 

x^-\-x*y7C-\-y-=a. 


J  2a 

*  J    '  On 


3  3 

.•.  x*—y* 


3  3 

x*+y* 


.     fib—a? 

~±V~2a~ 

,      /3a2— b 


'.  adding  and  subtracting,         xf       =A:U—% ±\/ 


36— a2 


2a 


3  /3a2— 6        /36-a* 


15) 


Hence  the  corresponding  values  of  a:  and  y  are 


J-  -/3a2— 6+  ■/36— as  )  s 

V8a 


x= 


±  -v/3a2— 6—  -v/3i— a2  >  if 


-I 


and 


-/8a 


-J-  -/3a2— /,>  —  V36— a2  if  ^  ±  -/3a2— 6+  -/3&— a2  ^  f 


-/8a  )  4  -/8a 

The  following  require  the  completion  of  the  square  : 


example  v. 


C  x+2/+2-2+7/2=a (1)  * 

I  x-y+x*-y*=b (2)  S 


Add  (1)  and  (2),  2x24-2.r=a+Z> (3) 

Subtract  (2)  from  (1),  2tf-\-2y=a  —  b     (4) 

Equations  (3)  and  (4)  are  common  adfected  quadratics;  solving  these  in  the 
usual  manner,  we  find 

— 1±  -v/l  +  2fl  +  2// 


x=- 


— 1±  yfl-\-2a— 26 


EXAMPLE    VI. 

$  x  +y  =    G (l)  > 

I  x"+i/4=272 (2)  S 

Raise  (1)  to  the  4th  power. 

x* + 4.r37/  -f  Qxhf + 4xf + yl = 1 296. 
But  from  (2),  x4  -\-y4=  272, 

Subtracting  4X3?/ + 6x2y2  -f-  4x?/3         =1024, 

or  2x?/(2x2+ 3x?/4- 2t/2)  =  1024 (3) 

But  by  (1),  2x!/(2x2+4x2/4-2?/2)  =  144xi?/ (4) 

Subtracting  (3)  from  (4),  ■2.ry:  =  liAxy— 1024. 


222 


ALGEBRA. 


Transposing  and  dividing  by  2, 

x-y-  — 72x1/  =—512. 

Completing  tl  e  square,  x"-y'i  — 72x_y-f  1296=1296— 512, 


or 


First,  let  us  suppose  xy  =  8. 

By(i), 

And 

Subtracting, 


(xy— 36)-=   784^ 

.-.  xy  —  3C)    =±  v/784 

xy  =36  ±28 

=64,  and  8. 

X'+2xy+f-  =  36, 

4xy         =32. 
x3 —  2xy-\-y-  =  4 

•.  x— ?/  =i2, 
x+y  =6. 


and 


But 

.-.  adding  and  subtracting, 

x=4  I 

y=2  $   """  }  !/  =  4 
Secondly,  et  us  take  the  other  value  of  xy,  or  64. 
By  (1),  x"-+2xy+y-=        36, 

4xy         =      256. 
Subtracting,  x- — 2xy  +  y-  =  —  220, 


But 

.•.  adding  and  subtracting, 


.-.  x— ?/  =±  V—  220. 
x+y  =6. 


x=- 


6+  V— 220" 


y=- 


6—  •/—220 


6— V— 220" 


and 


y=- 


6+  V— 220 


' 


Hence,  in  the  above  equations,  two  of  the  roots  of  x  and  j  are  pos*  -le,  and 
two  impossible. 

(7)*  2x  +  Sy  =118 (1)  > 

5*2— 7i/-=4333 (2)  \ 

x=35  )         x=—  229A> 

Ans.         ,_  >  and  .'.   > 

3/= 16  S         7/=      192/,  S 

(8)     8x+23y  =  2.t34.2y8 (1)  i 

341/4-   6.r°— 52/2=13x^/+24 (2)  \ 

—  181) 


Ans. 


x=3 

7/  =  2 


133 
34 


.'/  =  • 


X: 


55jpj/H14 

~~ 26 


y=- 


—  9±3-y/in4 


(9)  (x-y)(.i*-if)  =  a (1)  , 

CM-yK^+y9)^  (2)  s 


Ans.  x= 


V 26— a ±  Va  \' 26— a ?V'i 

-2/= 


2V26-a 


2V26— a 


•  The  following  examples,  though  a  valuable  .  are  likely  to  detain  the  stu-lont 

io"",  and  -.._■    be  omitted. 


QUADRATIC  EQUATIONS. 


223 


(10)    xyz 


x+y 
xyz 


=a 


2/+2 
xyz 


=  6 


x-{-z 


(11)  x+y=a, 
x*+y*  =  b. 
•-. — 96— x*y*, 


— 


(1) 
(2) 


(3) 
2abc(ab-\-bc — ac) 


^_v 


I         2abc(a 
"\J  (ab-\-ac  —  bc)(bc-\-ac — aby 
j        2abc(bc-\-ac —  ab) 
V=  ^y]{ab+ac— hc)(,ib  -{-be— ocj' 
I         2abc(ab-\-ac — be) 
~      \(ab-{-bc — ac)(bc-\-ac— ab)' 
a        Lb  — a3  a        lib— a3 

Ans.  x=4,  or  2,  or  3i  \/21» 
,  or  3  ^p  -/21. 

-  -/a2"— c2n)°, 


-  V«2n—  c20)"' 

or  —  3±  -/3. 

or  —  3^=  -v/3- 

'5,  ^=—3-1-1/7, 
^=4,  or  1. 


or 


—13-j- -/— 39 


or 


-13^1/— 39 


;  also,  .r=5,  or  -, 
:  5 

Iso,  g=3,  or  — 15. 


Ans.  x=9,  or 


3'=4'  or  7 


Ans.  x~5,  or 


25 

r 

17 

10' 


y=3,or  — . 


Ans.  £=G  or 


9 


y=12  or  — 9. 


^ 
^ 


-13-J-i/— 47 


-1-rt-l/— 11 


— 47        l^Sy^c 


-,  or 


or 


i±V-n 


222  ALGEBRA. 

Transposing  and  dividing  by  2, 

xhf—l  =—512. 

Completing  tl  e  square,  afy8— 72iy+ 1296=1296—512, 

or  (ry— 36):  =   784._ 

.-.  xy  —  36    =±  a/ 784 
ry  =36  ±28 

=61,  and  8. 

First,  let  us  suppose  xy  =  8. 

By  (1),  x*+2xy+f-=36, 

And  4xy         =32. 

Subtracting,  x*—2xy-\-y"  =  4 

•.  a:— y  =±2, 

But  .r+2/  =6* 
.-.  adding  and  subtracting, 


Secondly,  et  u 

By  (i). 

Subtracting, 

But 

.-.  adding  and  subt 


y 

Hence,  in  the  at 
two  impossible. 

(7)*  2x  +Sy  = 
5x2—7y"= 


(8)     8x-\-23y 
34i/4-   Gx*- 


(9)  (.r-7y)(.r°_:. 
(x+y)(z»+j 


x=i  I  and  $  a=2  } 


•  The  follow  i 
Irm  -. 


QUADRATIC  EQUATIONS. 


223 


(10)    xyz 


x+y 

xyz 


=a 


y+z 

x-\-z 


=b 


(1) 
(?) 

(3) 


111)  x+y  =a, 
x3-\-y3  =  b. 

(12)  4xy  =  9G  — x-y** 

x+y  =6. 

(13)  .Tn+7/"=2an, 


xy 


(14)  x2+x+y  =  18  — y\ 

xy=6. 

(15)  x»-f  2^+7/-+2x=120— 2^ 

(16)  afi+ft—x—i?=78, 

XV  +aH-y=39. 

(17)  aPy*—7xy*— 9-43=71;-, 

xy— y=l~- 
(is)  »— *Vxy+y— yx-\-Vy=o, 

Vx+Vy=5- 

,    ,   x*  ,  4a;     85 
a;—  2/  =2. 


i         2abc(ab-\-bc — ac) 

\{ab-\-ac — bc)(bc-\-ac — ab)' 

j        2abc(bc-\-ac — ab) 

■,==      -y (ab-{-ac  —  bc)(ab-fbc—acy 

J        2abc(ab-\-ac — be) 

=      \ \ab-{-bc—ac)(bc-{-ac—ab)' 

a        Ub—a?  a        lib— a3 

Ans.  a.ji^-jg-,  ^-^-^ 

Ans.  x=4,  or  2,  or  3±  \/21, 
2/=2,  or  4,  or  3=p  ■/21. 

Ans.  .r=(anIt  V«2"  — c2n)'a, 
c*  c2 

*      («nrt  V«2n—  c20)"' 
Ans.  .r=3,  or  2,  or  — 3±  V3, 
i/=2,  or  3,  or  — 3=f  V3- 

Ans.  a;= — JFFi/ 5,  #= — 3-J-i/5> 
also,  a:=6,  or  9,        y=4,  or  1. 

-13-tv/— 3» 


20    - /-: K/-T — =2. 

Vj;-J-y      V    3a: 

(21)  *<— 2«,ty-r-y2=49 


**.  -2j  ■  -  y  - -j- y4— x-+i/-=i20. 


Ans.  a;=9,  or  3,  or 


Ans.  x- 


y=3,  or  9,  or 
—19 


— 13=PV— 39 


1 


-;  also,  x^=5,  or  -, 

17^6^—2  5 

y= — GjL-v/ — 2;  also,  #=3,  or  — 15. 

25 

Ans.  a:=9,  or  — , 
4 


y— 4> or  7-  * 

17 
Ans.  1=15,  or  — , 

—3 
y=3,  or  — . 


Ans.  x=6  or 


10 

9 


#=12  or  — 9. 
Ans.  .r=-|-3,  or  J^lA'  or  ir\/— — ^r 


3-J-i/ — 47 


/l.'>4-3i/5  1      /" 


w=y^-±J=HJg^ 


l4--j/— 47        l4-3i/5 
2,  or  — 1,  or  -Si- ,  or  ■      „    -, 

i±!A=n 


or 


224  ALGEBRA. 

(22)  xi/-\-xy"-=12, 

x-\-xyt=18. 

(23)  x— x*=3— y, 
4— x  =y—y. 


Ans. 

x  = 

"-, 

or  16, 

:■'- 

=■-', 

1 

Ans 

.  xz 

=4, 

1 

9= 

=  1. 

9 

or  -. 

4 

1        —974-^/6045 
(24)  (x2+%=x  y  +12t>,  Ans.  *=5,  or  -■  or 3=J: , 


(x*-\-\)y=xiy* — 744.  y=*  or  150,  or 


58 


r-v/0045 

(25)  x  -\-y  4-V^+y=12,  Ans.  x=5,  or  4, 
a<s-|-y:)=189.  y=4,  or  5. 

(26)  x^y^+x— y=132,  Ans.  x=ll,  or  — 1,  or  61±i/— 3716, 
(I2-f-3,:)(cc_3,)=1220.  y=l,  or  ~ n.  or  —  6lT" y/— 371C. 

>;S")  x^=2y?,  Aqs.  x=14|3,  or  8, 

8x*— y*=14.  y=ys"','".  or  4. 

^28)  x*-\- y%=3x  (see  note,  page  217),  Ans.  x=4,  or  1, 

xl~-\-y^—x.  y=8. 

(29)  ;c-fa;i=:^ip^+4.  Ans.  *=4,  or  1, 

y+«y=y3+4y- 5'=1'  or  — 2- 

(30)  2x+y=26— 7v^H^+4,  Ans.  x=2,  or  —10, 
2x — \/y      15     2.r-f-i/y 


y=256.  or  256]*. 


131) 8^/x— 9x^=9^— 16xy,  Ans.  x=t, 

5x=4+25y:.  y=±? 

o 

(3f )   -Cx— y^=6_yM,  Ans.  x=4,  or  16, 

ar»      12       x 

<B3)     /5i/x+5-i/^+-/y=10— \/*.  Ans.  x=9,  or  4. 

•V/x5+-v/y6=275.  ^=4,  or  9. 

PROBLEMS  PRODUCING  PURE  EQUATIONS. 

(1)  What  two  numbers  are  those  whose  sum  is  to  th  r  as  10  to  7 
and  whose  sum,  multiplied  by  the  less,  produces  270  ? 

Ans.  ±21  and  ±9 

(2)  There  are  two  numbers  in  the  proportion  of  4  to  5,  and  the  difference 
of  whose  squares  is  81.     What  are  tho  numbers? 

Ans.  ±12  and  ±15. 

(3)  A  detachment  from  an  army  was  marching  in  regular  column,  with  5 
men  nunc  in  depth  than  in  front;  but  upon  the  enemy  coming  in  Bight,  the 
front  was  increased  by  845  men,  and  by  this  movement  the  detachment  was 
drawn  up  in  five  lines.    Required  the  number  of  men  ? 

.  4550. 

(4)  Two  workmen,  \  and  B,  wen  •  1  to  work  for  a  certain  number 
of  day  ■  sit  different  rates.     At  the  end  of  the  time,  A,  who  hud  been  idle  4  of 


QUADRATIC  EQUATIONS.  225 

those  day;,  had  75  shillings  to  receive ;  but  B,  who  had  been  idle  7  of  those 
days,  received  only  48  shillings.  Now,  had  B  been  idle  only  4  days  and  A  7, 
they  would  have  received  exactly  alike.  For  how  many  days  were  they  en- 
gaged, how  many  did  each  work,  and  what  had  each  per  day  ? 

Ans.  A  worked  15  and  B  12  days. 

A  received  5  and  B  4  shillings  per  day. 

(5)  A  vintner  draws  a  certain  quantity  of  wine  out  of  a  full  vessel  that  holds 
256  gallons,  and  then  filling  the  vessel  with  water,  draws  off  the  same  quantity 
of  liquid  as  before,  and  so  on  for  four  draughts,  when  there  were  only  81 
gallons  of  pure  wine  left.     How  much  wine  did  he  draw  each  time  ? 

Ans.  64,  48,  36,  and  27  gallons* 

PROBLEMS  WHICH  PRODUCE  AD'FECTED  OR  COMPLETE  QUADRATIC 

EQUATIONS. 

'  PROBLEM  1. 

190.  To  find       -.umber  such  that  twice  its  square,  augmented  by  three 

times  the  numbeV,  is  equal  to  65. 

Let  x  be  the  number  required,  we  have  for  the  equation  of  the  problem, 

2z2+3x=65. 

3        /65      9~  3     23 

Solving  the  equation,  x=  — --j-^— +— =  —  -  J-—. 

13 
Hence  x=5 ;  x=  — — . 

The  first  of  these  two  values  satisfies  the  conditions  of  the  problem,  as  stated 
in  the  enunciation  ;  for,  in  fact, 

2(5)2+3X  5=2x25+15 
=65. 

In  order  to  interpret  the  meaning  of  the  second  value,  let  us  observe,  that 
if  we  substitute  — x  for  -f-x  in  the  equation  2x2-f-3x=65,  the  coefficient  of  3x 
alone  will  change  its  sign,  for  ( — x)2=(+x)2=x2.  Hence  the  value  of  x  will 
no  longer  be 


■6* 


3  ,  23 

3     23 

but  will  become  x=  +  -  ±  — . 

1  4      4 

13 
Hence  x=— • ;  x= — 5, 

where  the  values  of  x  differ  from  those  already  found  in  sign  alone. 

13 
Hence  we  may  conclude  that  the  negative  solution  — — ,  considered  with- 
out reference  to  its  sign,  is  the  solution  of  the  following  problem : 

To  find  a  number  such  that  twice  its  square,  diminished  by  three  times  the 
number,  is  equal  to  65. 
In  fact,  we  have 

/13\*  13_169     39 

2V2/  ~3X"2_=_2""""2" 
=  65. 
P 


226  ALGEBRA. 

PROBLEM  2. 

A  tailor  bought  a  certain  number  of  yards  of  cloth  for  j  2  pounds.  If  he  had 
paid  the  same  sum  for  3  yards  less  of  the  same  cloth,  then  the  cloth  would 
have  cost  4  shillings  a  yard  more.     Required  the  number  of  yards  purchased. 

Let  x  be  the  number  of  yards  purchased. 

240 
Then is  the  price  of  one  yard,  expressed  in  shillings. 

If  he  had  paid  the  same  sum. for  3  yards  less,  in  that  case  the  price  of  eacti 

240 
would  be  represented  by -. 

X^—  o 

iJut  by  the  conditions  of  the  problem,  this  last  price  is  greater  than  the 
former  by  4  shillings ;  hence  the  equation  of  the  problem  will  be 

240       240 


x-3-  x  +4' 
or  x2— 3x=180. 

Whence  a:=-_L_^_+i8n=5_.fc._ 

.•.  x=15  ;  x=  — 12. 

The  value  of  x=15  satisfies  the  conditions  of  the  problem,  for 

240  240 

-=lG;-=20, 

the  price  of  each  yard  in  the  first  case  being  1G  shillings,  and  in  the  last  case 
20,  which  exceeds  the  former  by  4  shillings. 

With  regard  to  the  second  solution,  we  can  form  a  new  enunciation  to  which 
it  will  correspond.  Resuming  the  original  equation,  and  changing  x  into  — x, 
it  becomes 

240  240 


i=— :+4i 


— x — 3      — x 
or 

240       240 


•4, 


x+3        x 

an  equation  which  may  be  considered  as  the  algebraic  representation  of  the 
following  problem  : 

A  tailor  bought  a  certain  number  of  yards  of  cloth  for  12  pounds.    If  he  had 
paid  the  same  sum  for  3  yards  inore,  then  the  clot li  would  have  cost  4  shillings 
a  yard  less.     Required  the  number  of  yards  purchased. 
The  above  equation  when  reduced  becomes 

x*+3x=180, 
instead  of  x' — 3x=180,  as  in  the  former  case;  solving  tho  above,  we  find 

x=rl2;  x=  — 15. 
The  two  preceding  problems  illustrate  tho  pr  aiciplo  explained  with  regard 
to  problems  of  the  first  degree. 

PROBLEM    3. 

A  merchant  purchased  two  bills;  one  for  88770,  payable  in  9  months,  the 
other  for  $7-lH-t,  puyublo  in  8  months.  Pot  the  firsl  lie  paid  (1200  more 
than  for  tho  second.     Required  the  rate  of  interest  allowed. 


aUADRATIC  EQUATIONS.  227 

Let  x  represent  the  interest  of  $100  for  1  month. 

Then  12x,  9x,  8x  severally  represent  the  interest  of  $100  for  1  year,  9 
months,  8  months. 

And  100  +  9x,  100+8r  represent  what  a  capital  of  $100  will  become  at 
the  end  of  9  and  of  8  months  respectively. 

Hence,  in  order  to  determine  the  actual  value  of  the  two  bills,  we  have  the 
following  proportions : 

8776X100 
100  +  9j:100:;8776:'100+9;c 

7488X100 
100+8ar;100;:7488:    100,8x  • 

The  fourth  terms  of  the  above  proportions  express  the  sum  paid  by  the 
merchant  for  each  of  the  bills. 

Hence,  by  the  conditions  of  the  problem, 

877600        748800 
100+9x*~100  +  8x— 12     ' 
or,  dividing  each  member  by  400, 

2194  1872     _ 

100  +  9x-— 100  +  8x— 
Clearing  of  fractions  and  reducing, 

216a:2+4396x=2200. 

Whence 

2198        /2200      /2198\a 
x—~  216.      V  216  "H2I6/ 


— 2198J;  V 5306404 
2l6 


— 2198±  -v/5306404 

•  12x - 

•  l4X—  18 

— 2198±2303.5 


18 

.-.  12x=5.86 ;  and  12x=  —  250.08 

The  positive  solution,  12x=5.86 ,  represents  the  required  rate  of  in- 
terest per  cent,  per  annum. 

"With  regard  to  the  negative  solution,  it  can  only  be  considered  as  connected 
with  the  other  by  the  same  equation  of  the  second  degree.  If  we  resume 
the  original  equation,  and  substitute  — x  for  +.r,  we  shall  find  great  difficulty 
in  reconciling  this  new  equation  with  an  enunciation  analogous  to  that  of  the 
proposed  problem. 

problem  4. 

A  man  purchased  a  horse,  which  he  afterward  sold  to  disadvantage  for  24 
pounds.     His  loss  per  cent,  by  this  bargain,  upon  the  original  price  of  the 
horse,  is  expressed  by  the  number  of  pounds  which  he  paid  for  the  horse 
Required  the  original  price. 

Let  x  be  the  number  of  pounds  which  he  paid  for  the  horse. 

Then  x — 24  will  represent  his  loss ; 
But,  by  the  conditions  of  the  problem,  his  loss  per  cent,  is  represented  by  the 
number  of  units  in  x ; 


228  ALGEBRA. 

x 
His  loss  per  cent,  on  one  pound  is  — -. 

x* 
.«.  his  loss  per  cent,  on  X  pounds  must  be  — ,  or  x  times  as  greut. 

T'his  gives  the  equation, 

x3 

=  r 24 

100 

x=50±  Vl00=50±10. 

Hence  x=60  ;  x=40. 

Both  these  solutions  equally  fulfill  the  conditions  of  the  problem. 

Let  us  suppose,  in  the  first  place,  that  he  paid  60  pounds  for  the  horse,  since 

be  sold  it  for  24,  his  loss  was  36.     On  the  other  hand,  by  the  enunciation,  his 

60  60X60 

'oss  was  60  per  cent,  on  the  original  price;  i.  c,  r-—  of  60,  or  =36  j 

thus  60  satisfies  the  conditions. 

In  the  second  place,  let  us  suppose  that  he  paid  40  pounds ;  his  loss  in  this 

case  was  16.     On  the  other  hand,  his  loss  ought  to  be  40  per  cent,  on  the 

40  40X40 

.riginal  price  ;  i.  c,  r-nr  of  40,  or  =16  ;  thus  40  also  satisfies  the  con 

ditions. 

GENERAL  DISCUSSION  OF  THE  EQUATION  OF  THE  SECOND  DEGREE 

191.  The  general  form  of  the  equation,  the  coefficients  being  considered  ir- 
pendently  of  their  signs,  is 

x*-\-px-\-q~0. 

P% 
I.,  II.  Let  q  be  positive  and  <  — , 

!P      Ip% 
I.  Ifp  be  positive,  x= —  -±W— — q,  and  both  values  are  negative.* 

P        fp* 
II.  Ifp  be  negative,  x=-J--±W— — q,  and  both  values  are  positive. 


P* 
TU.,  IV.  Let  q  be  positive  and  >  — , 


(  P        fW      ' 

III.  If  j?  be  positive,  z=  —  -±y  — — q, 

ns — 

p      IP' 

IV.  U]}  benegative,  x=+— ±y  — — 7, 


and  both  values  are  imagi- 
nary, f 


•  In  this  and  all  the  following  values  of  x,  calling  the  term  -  before  the  radical  the  ra- 
tional part,  and  -  /— -^q  the  radical  part,  we  perceive  that,  when  q  is  positive,  the  radical 

part  is  greater  than  the  rational,  since  ..  J—  alone  equals  ;,  the  rational  part  1  and  tli 

of  the  wholo  expression  is  that  of  the  radical  part ;  but  when  q  is  negative,  the  radical 
part  is  less  than  the  rational,  and  the  sign  of  the  wholo  expression  is  that  of  the  rational 
part. 

t  In  this  case,  if  wo  examino  the  general  equation,  we  shnll  Bad  that  the  conditions  ora 
abaurd  ;  for,  transposing  q,  and  completing  the  square,  wo  havo 


QUADRATIC  EQUATIONS. 

ml 


229 


V.,  VI.  Let  q  be  negative  and  <  — , 

P        Fp1, 

V.  Up  be  positive,  x=  —  -±w  — -  +  <7» 

VI.  If_p  be  negative,  .r=  +  -±->y— +  ?, 
VII.,  VIII.  Let  q  be  negative  and  >-r, 

w  In5 

VII.  Up  be  positive,  z=—  ^iy^+tf' 

VIII.  Ifp  be  negative,  ar=+!±^/j+?, 


and  one  value  is  positive 
the  other  negative. 


IX.,  X.  Let  q=—,  and  be  positive. 


IX.  If  p  be  positive,  x=  — 

X.  If p  be  negative,  x=-\- 
XI.,  XII.  Let  g=0, 


•  and  the  two  values  are  equal. 


P  ,  P 


XI.     If  jp  be  positive,  .r=— -±^,  one  value  =— p,  the  other  =0. 


2"1"2 

P,P 
2±2 


XII.  Up  be  negative,  a:=+-±^,  one  value  =+p,  the  other  =0. 


XIII.  Let  q  be  negative. 

{XIII.  p=0,  x=±  V?!  the  two  values  are  equal  with  opposite  signa 

XIV.  Let  q  be  positive, 

{XIV.  p=0,  x=Az  V — q,  both  values  are  imaginary. 

XV.  Let  q  =  0, 

{XV.  j>=0,  then  x=0,  or  both  values  are  equal  to  0. 


P1 


but  since  — — q  is,  by  hypothesis,  a  negative  quantity,  we  may  represent  it  by  — to,  where 
m  is  some  positive  quantity  ;  then 

x3rt  px-\-—= — to 


(*±?)+»=o; 


that  is,  the  sum  of  two  quantities,  each  of  which  is  essentially  positive,  is  equal  to  0,  • 
manifest  absurdity.    Solving  the  equation, 


-P 


and  die  symbol  \/ — to,  which  denotes  absurdity,  serves  to  distinguish  this  case.     Hence, 
when  the  roots' are  imaginary,  the  problem  to  which  the  equation  corresponds  is  absurd. 

We  still  say,  however,  that  the  equation  has  two  roots  ;  for,  subjecting  these  values  of 
x  to  the  same  calculations  as  if  they  were  real,  that  is,  substituting  them  for  x  in  the  pro> 
posed  equations,  we  shall  find  that  they  render  the  two  members  identical 


J30  ALGEBRA. 

XVI.  One  case,  attended  with  remarkable  circumstances,  stl  remains  to  be 
examined.     Let  us  take  the  equation 

a3?-\-bx — c=0. 

Whence  x= . 

2a 

Let  us  suppose  that,  in  accordance  with  a  particular  hypothesis  made  on  the 

given  quantities  in  the  equation,  we  have  a  =  0;  the  expression  for  x  then 

becomes 

f 


_&±6 
x= — ;  whence 


X=0 


0 
0 
—2b 


0 

The  second  of  the  above  values  is  under  the  form  of  infinity,  and  may  be  con 

sidered  as  an  answer,  if  the  problem  proposed  be  such  as  to  admit  of  infinite 

solutions. 

0 
We  must  endeavor  to  interpret  the  meaning  of  the  first,  -. 

In  the  first  place,  if  we  return  to  the  equation  ax"-\-bx — c=0,  we  perceive 

that  the  hypothesis  a  =  0  reduces  it  to  bx=c,  whence  we  derive  x=-r,  a.  finite 

and  determinate  expression,  which  must  be  considered  as  representing  the  true 

value  of  -  in  the  case  before  us. 

That  no  doubt  may  remain  on  this  subject,  let  us  assume  the  equation 

ax"1  -\-bx — c=0, 

and  put  a.'=-,  the  expression  will  then  become 

a       b 

-+  -— c=0. 

y^  y 

Whence  cy"1 — by — a=0. 

Let  a  =  0,  this  last  equation  will  become 

cy"—by=0, 

from  which  we  have  the  two  values  y=0,  y=- ;  substituting  these  values  in 
jr=-,  we  deduce 

y 

1  c 

1°.  x=-  ;  2°.  x=t-* 


"  To  show  more  distinctly  how  the  indeterminate  form  arises,  let  us  resume  the  general 
value  of  one  of  the  roots. 


— b+\/b°~{-iac 
2a 
If  a  were  a  factor  of  both  the  numerator  and  denominator,  it  might  bo  supprossed,  and 
then  a,  being  put  equal  to  zero,  would  give  the  true  value  of  x.    We  can  not,  indeed, 
■how  the  existence  of  this  factor  in  the  two  terms  of  the  fraction  as  it  stands  ;  but  if  wa 


multiply  both  numerator  and  denominator  by  — b — \/b--\-\ac,  it  becomes 

_  (~ Z>-fVfr»+4ac)(— b— y/b*+i<t, ) 
—2a{b+-i/b:+4ac) 


QUADRATIC  EQ.UATI0N6.  231 

—  26 
"With  respect  to  the  value  x=         ,  it  is  only  to  be  observed  that  the 

divisor  zero,  having  to  be  regarded  as  the  limit  of  decreasing  magnitudes,  either 
positive  or  negative,  it  follows  that  the  infinite  value  ought  to  have  the  am- 
biguous sign  i. 
Thus  the  values  of  r,  to  recapitulate,  become 

c  , 

X=y,  X=±CO. 

It  is  remarkable  that,  for  this  particular  case,  we  have  three  values  of  a. 
while  in  the  general  case  there  are  but  two. 

To  comprehend  how  these  values  truly  belong  to  the  equation  ax2-\-  bx 
— c=0,  put  it  under  the  form 

—  bx-\-c 

^ =a. 

x2 

— bx-\-c 
When  a  =  0,  the  question  is  to  find  values  which  will  render —  zero 

c 
We  see  that  x=r  will  do  it;  and  as  the  same  expression  can  be  written  under 

b      c 
the  form  — -+"^  we  perceive  that  it  becomes  zero  also,  from  the  value9 

X=±aQ.* 

XVII.  Let  us  cousider  the  still  more  particular  case  still,  where  we  have, 

0 

at  the  same  time,  a=0,  6  =  0.     Then  the  two  general  values  of  x  become  -. 

We  have  seen  above  that  the  first  may  be  changed  into 

2c 

6+  A/6'-+4ac' 
Transforming  the  second  in  a  similar  manner,  it  becomes 


( — 6 —  V63+4ac)(  —  6+  V*s+4ac)  —2c 

X= 


2a(— 6+  V62+4ac).  —6+  *Jb2+4ac 

In  which,  making  a=0,  6  =  0,  the  values  of  x,  thus  transformed,  both  give 
r=oo  ;  and  here,  also,  the  infinity  ought  to  be  taken  with  the  sign  i. 

If  we  suppose  a  =  0,  6=0,  <?=0,  the  proposed  equation  will  become  alto- 
gether indeterminate. 

The  numerator,  being  the  product  of  the  sum  and  difference  of  two  quantities,  is  equal 
to  the  difference  of  their  squares,  to  wit :  b2 — (6*-j-4ac)= — 4ac.  We  see,  therefore,  that 
2a  is  a  common  factor  to  the  numerator  and  denominator  of  the  last  expression.  Suppress- 
ing it,  we  have 

2c 


b-\-\Zb*-{-4ac 
in  which,  if  we  make  #=0,  it  gives  «=;. 

*  In  the  analytic  theory  of  curves  these  values  answer  to  the  intersections  of  the  axis 
of  abscissas  with  the  curve  of  the  3°  order,  the  equation  of  which  is  yx-\-bx-\-c=0.  If  this 
curve  be  constructed,  it  will  be  found  to  cut  the  axis  of  abscissas  first  at  a  finite  distance 
from  the  origin,  and  besides  has  this  axis  for  an  asymptote  both  on  the  side  of  the  positive 
and  negative  abscissas,  which  amounts  to  saying  that  it  cuts  it  at  infinity  in  either  di- 
rection. 


232  ALGEBRA. 

192.  Let  us  uow  proceed  to  illustrate  the  principles  established  in  this  gen 
eral  discussion,  by  applying  them  to  different  problems. 

PROBLEM    5. 

To  find  in  a  line,  A  B,  which  joins  two  lights  of  different  intensities,  a  point 
which  is  illuminated  equally  by  each. 


P3  A  Pi        B  1\. 

(It  is  a  principle  in  Optics  that  the  intensities  of  the  same  light  at  different 
distances  are  inversely  as  the  squares  of  the  distances.) 

Let  a  be  the  distance  A  B  between  the  two  lights. 

Let  b  be  the  intensity  of  the  light  A  at  the  distance  of  one  foot  from  A. 

Let  c  be  the  intensity  of  the  light  B  at  the  distance  of  one  foot  from  B. 

Let  P ,  be  the  point  required. 

LetAP,=z;  .-.  BP,=a- x. 

By  the  optical  principle  above  enunciated,  since  the  intensity  of  A  at  the 

distance  of  1  foot  is  b,  its  intensity  at  the  distance  of  2,  3,  4, feet  must  be 

b    b     b  b 

-,-,—;  hence  the  intensity  of  A  at  the  distance  of  x  feet  must  be  —.     In  the 
4   9    lb  J  x* 

c 
same  manner,  the  intensity  of  B  at  the  distance  a — x  must  bo  -. ;  but 

according  to  the  conditions  of  the  question,  these  two  intensities  are  equal ; 
hence  we  have  for  the  equation  of  the  problem 

b_    _c_ 

x*~(a^xy-' 
Solving  this  equation,  and  reducing  the  result  to  its  most  simple  form, 

a  y/b 

X~  Vb±Vc 
We  shall  now  proceed  to  discuss  these  two  values  : 

a  V 'c 


a  \/b 

1° .1 


a\Jb 
2° x  =  - 


Vb-Vc) 


a — xz 


^6+/c     whence  VH-/< 


{a-r=Vi 


— a  y/c 
b^Wc 


I.  Let  6>c. 

ay/b  y/b 

The  first  value  of  x,     . ,— , — 7-,  is  positive,  and  less  than  a,  for  — 77—; 

J  yb-\-yc        '  v'»+vc 

is  a  proper  fraction;  hence  this  value  gives  for  the  point  equally  illuminated  a 

point  Pj,  situated  between  the  points  A  and  B.     Wo  perceive,  moreover,  that 

the  point  P,  is  nearer  to  B  than  to  A;  for,  since  l^>r,  we  have 

Vl>+  V>>>  V'+  Vc,  or  2  Vb>  V^+  V<-.  and  .-.   j*  >g, 

(iy/b  a 

and.  consequently,     . .         ,   >;,-      1  his  is  manifestly  the  result  at  which  we 

ought  to  arrive,  for  we  hero  suppose  the  intensity  of  A  to  be  greater  than  that 

ofB. 

.  a  y/c  a 

The  corresponding  value  of  a — x,     . .  . — j-,  is  positive,  and  less  than  -. 

"  V  b 
Die  second  value  of  r,  —jt r-.  is  positive,  and  greater  than  a,  for 


QUADRATIC  EQUATIONS.  233 

Vb>  Vb-  Vc,  •••  Vb_Vc>h  and  ...  -±—->a. 

This  second  value  gives  a  point  P2,  situated  in  the  production  of  A  B,  and  to 

the  right  of  the  two  lights.     In  fact,  we  suppose  that  the  two  lights  give  forth 

rays  in  all  directions  ;  there  may,  therefore,  be  a  point  in  the  production  of  A  B 

equally  illuminated  by  each,  but  this  point  must  be  situated  in  the  production 

of  A  B  to  the  right,  in  order  that  it  may  bo  nearer  to  the  less  powerful  of  the 

two  lights. 

It  is  easy  to  perceive  why  the  two  values  thus  obtained  are  connected  by 

the  same  equation.     If,  instead  of  assuming  A  Pt  for  the  unknown  quantity  x, 

b            c 
we  take  A  P2,  then  B  P2=a- — a,  thus  we  have  the  equation  -£=-. rj ;  but 

since  (x — a)3  is  identical  with  (a — x)2,  the  new  equation  is  the  samo  as  that 
already  established,  and  which,  consequently,  ought  to  give  AP,  as  well  as 
AP,. 

The  second  value  of  a — x,  —rr j~,  is  negative,  as  it  ought  to  be,  being 

estimated  in  a  contrary  direction  from  the  first,  on  the  general  principle  already 
established,  that  quantities  estimated  in  a  contrary  sense  should  be  represented 

with  contrary  signs  ;  but  changing  the  signs  of  the  equation  a — x=  -jr — -7-, 

y  b  —  v  c 

a  V '  c 
we  find  x — a=    ., .  ,  and  this  value  of  x — a  represents  the  absolute 

length  of  B  P3. 

II.  Let  b  <c. 

The  first  value  of  x,     .,     — y-  is  positive,  and  less  than  -,  for  -\/b-\-  Vc 

>  *"+  ^  -  *»+  *>■/*  -  -jwtA  -  iwjsl- 

a  y  c  d 

The  corresponding  value  of  a — .r,      .,         .  ,  is  positive,  and  greater  than  - 

Hence  the  point  P  l  is  situated  between  the  points  A  and  B,  and  is  nearer 
to  A  than  to  B.  This  is  manifestly  the  true  result,  for  the  present  hypothesis 
supposes  that  the  intensity  of  B  is  greater  than  the  intensity  of  A. 

ay/b              — a-y/b     . 
The  second  value  of  x,  —j-r j-,  or  —. jr,  is  essentially  negative.     In 

order  to  interpret  the  signification  of  this  result,  let  us  resume  the  original 

,        ,  b  c 

equation,  and  substitute  — x  for  +x,  it  thus  becomes  —=7 — ; — ^.     But  since 
1  '  x-     (a-j-.r)- 

(a — x)  expresses  in  the  first  instance  the  distance  of  B  from  the  point  required, 
a-\-r  ought  still  to  express  the  same  distance,  and,  therefore,  the  point  re- 
quired must  be  situated  to  the  left  of  A,  in  P3,  for  example.  In  fact,  since 
the  intensity  of  the  light  B  is,  under  the  present  hypothesis,  greater  than  the 
intensity  of  A,  the  point  required  must  be  nearer  to  A  than  to  B. 

— a  V ' c               a  -\/c 
The  corresponding  value  of  a — .r,  — ry j-,  or     .   .,,  is  positive,  and 

the  reason  of  this  is,  that  x  being  negat  ve,  a — .t  expresses,  in  reality,  an 
arithmetical  sum. 


234  ALGEBRA. 

III.  Let  b=c. 

a 
The  first  two  values    of  x  and  of  a—x  are  reduced  to  -,  fvhich  gives  the 

bisection  of  A  B  for  the  point  equally  illuminated  by  each  light,  a  result  which 
is  manifestly  true,  upon  the  supposition  that  the  intensity  of  the  two  lights  is 
the  same. 

a  \/b 
The  other  two  values  are  reduced  to  — — ,  that  is,  they  become  infinite, 

that  is  to  sav,  the  second  point  equally  illuminated  is  situated  at  a  distance 
from  tho  points  A  and  B  greater  than  any  which  can  be  assigned.  This  re- 
sult perfectly  corresponds  with  the  present  hypothesis;  for  if  we  suppose 
the  difference  b—c,  without  vanishing  altogether,  to  be  exceedingly  small,  the 
second  point  equally  illuminated,  exists,  but  at  a  great  distance  from  the  two 

a  Vb 
lights  ,  this  is  indicated  by  the  expression     . .  _—r-,  the  denominator  of  which 

is  exceedingly  small  in  comparison  with  the  numerator  if  we  suppose  b  very 
nearly  equal  to  c.  In  the  extreme  case,  when  b=c,  or  V^ —  -/c=0,  the 
point  required  no  longer  exists,  or  is  situated  at  an  infinite  distance. 

IV.  Let  b=c  and  a=0. 
The  first  system  of  values  of  x  and  a—x  iu  this  case  become  0,  and  the 

second  system  -.     This  last  result  is  here  the  symbol  of  indetermination ;  for 

if  we  recur  to  the  equation  of  the  problem 

b__        c 
x*~  (a— x)4' 

or 

(b—c)z~—2abx=—a% 

it  becomes,  under  the  present  hypothesis, 

O.z2— 0..r=0, 
an  equation  which  can  be  satisfied  by  the  substitution  of  any  number  whatever 
for  x.     In  fact,  since  the  two  lights  are  supposed  to  be  equal  in  intensity,  and 
to  be  placed  at  the  same  point,  fliey  must  illuminate  every  point  in  the  line 
A  B  equally. 

The  solution  0,  given  by  the  first  system,  is  one  of  those  solutions,  infinite 
in  number,  of  which  the  problem  in  this  case  is  susceptible. 

V.  Let  a=0,  6  not  being  ==r. 
Each  of  the  two  systems  in  this  case  is  reduced  to  0,  which  proves  that  in 
this  case  there  is  only  one  point  equally  illuminated,  viz.,  the  point  in  which 
the  two  U glits  are  placed. 

Tho  above  discussion  affords  an  example  of  the  precision  with  which  algebra 
answers  to  all  tho  circumstances  included  in  the  enunciation  of  a  problem. 

■We  shall  conclude  this  subject  by  solving  one  <>r  two  problems  which  re 
quire  the  introduction  of  more  than  one  unknown  quantity. 

mom, km  6. 

To  find  two  numbers  such  that,  when  multiplied  by  the  numbers  a  and  b 
respectively,  the  sum  of  the  products  may  be  equal  to  2&,  and  the  product  of 
the  two  numbers  equal  top. 


aUADRATIC  EQUATIONS.  035 

Let  x  and  y  bo  the  two  numbers  sought,  the  equal  ons  of  the  problem  will 
be 

ax-\-by  =  2s (1) 

X1J=  P (2) 

From(l) 

2s  —  ax 

Substituting  this  value  in  (2)  and  reducing,  we  have 

ax2 — 2sx-{-bp=z0. 
Whence 

And  ••. 

s      1    , 

The  problem  is,  we  perceive,  susceptible  of  two  direct  solutions,  for  s  is 
manifestly  >  \/s'2 — a?bp ;  but  in  order  that  these  solutions  may  be  real  wo 
must  have  s2>,  or  =a2bp. 

Let  a  =  b  =  l ;  in  this  case  the  values  of  x  and  y  are  reduced  to 

£=s±  Vs"—  p,  2/=sT  V$~—p- 

Here  we  perceive  that  the  two  values  of  y  are  equal  to  those  of  x  taken  in 
an  inverse  order  ;  that  is  to  say,  if  s-f-  V '  s" — p  represent  the  value  of  x,  then 
* —  Vs'2 — P  will  represent  the  corresponding  value  of  y,  and  reciprocally. 

We  explain  this  circumstance  by  observing  that,  in  this  particular  case,  the 
equations  of  the  problem  are  reduced  to  x-\-y=2s,  xy=p,  and  the  question 
then  becomes,  Required  two  numbers  whose  sum  is  2s,  and  whose  product  is 
p,  or,  in  other  words,  To  divide  a  number  2s  into  two  parts,  such  Uiat  their 
product  may  be  equal  to  p. 

PROBLEM  7. 

To  find  four  numbers  in  proportion,  the  sum  of  the  extremes  being  2s,  the 
sum  of  the  means  2s',  and  the  sum  of  the  squares  of  the  four  terms  4c2. 

Let  a,  x,  y,  z  represent  the  four  terms  of  the  proportion ;  by  the  conditions 
of  the  question,  and  the  fundamental  property  of  proportions,  we  shall  have  as 
the  equations  of  the  problem 

a+z=2s (1) 

x-\-y  =  2s' (2) 

xy=a* (3) 

c?-{-a?4-y,4-z2=4c3 (4) 

Squaring  (1)  and  (2)  and  adding  the  results, 

a2 + .r2 + 7/ + z2 -f 2az + 2xy = 4  (s2 + s'2) . 
But  by  (4),  a2-j-;r2+?/2+z3  =4c-. 

Subtracting,  2az + 2xy  =  4  (s2 + s'2 — c9) . 

.-.  by  (3),  4az=4(s-+s'i—c°-)=4ty  .      (5) 

Squaring  (1),  a2+2«z  +  c-=4s2. 

But  by  (5),  Aaz        =4(s2+s'2— <:'). 

Subtracting,  a*—2az-{-zr=^A{c°r—s'-). 

Extracting  the  root,  a— z=  ± 2  y/c2 — *'s. 

But  by  (1),  a+z  =  2s. 


236  ALGEBRA. 


.•.  adding  and  subtracting,  a=s^z  \AC — s'J 

Precisely  in  the  same  manner  we  shall  find 


X  =  S';±:  -y/c"- 


y=s'=f  y/<?—#. 


The  four  numbers  will  therefore  be 


a=s+  Vc2— s'2,  x=s'+  j<? 


zz=s —  V ' c" — s's,  y=s' —  -/c9 — s5. 
These  four  numbers  constitute  a  proportion,  for  we  have 

az  =  (s  +  yfc"-  —  s"2){s  —  Vc-— s"-)= s2— c2+s' 

:H/  =  (s'  +  V^-^jfs'-  -v/c2  — S2)=5'2  — C'  +  S' 


9 


(8)  What  two  numbers  are  those  whose  sum  is  20,  and  their  product  36  ? 

Ans.  2  and  18. 

(9)  To  divide  the  number  60  into  two  such  parts  that  their  product  may 
be  to  the  sum  of  their  squares  in  the  ratio  of  2  to  5. 

Ans.  20  and  40. 

(10)  The  difference  of  two  numbers  is  3,  and  the  difference  of  their  cubes 
is  117.     What  are  those  numbers  ? 

Ans.  2  and  5. 

(11)  A  company  at  a  tavern  had  d£8  15s.  to  pay  for  their  reckoning;  but, 
before  the  bill  was  settled,  two  of  them  left  the  room,  and  then  those  who  re- 
mained had  105.  apiece  more  to  pay  than  before.     How  many  were  there  in 

company  ? 

Ans.  7. 

(12)  A  grazier  bought  as  many  sheep  as  cost  him  <£60,  and  after  reserving 
15  out  of  the  number,  he  sold  the  remainder  for  <£54,  and  gained  2s.  a  head  by 
them.     How  many  sheep  did  he  buy  ? 

Ans.  75. 

(13)  There  are  two  numbers  whose  difference  is  15,  and  half  their  product 
is  equal  to  the  cube  of  the  lesser  number.     What  are  those  numbers  ? 

Ans.  3  and  18 

(14)  A  person  bought  cloth  for  <£33  15s.,  which  he  sold  again  at  £2  B».  per 
piece,  and  gained  by  the  bargain  as  much  as  one  piece  cost  him.  Required  the 
number  of  pieces. 

\  us.  15. 

(15)  What  number  is  that,  which  when  divided  by  the  product  of  its  two 
digits,  the  quotient  is  3;  and  if  18  more  be  added  to  it,  the  digits  will  bo 
transposed ? 

Ans.  24. 

(1G)  What  two  numbers  are  those  whose  sum,  multiplied  by  the  greater, 
is  equal  to  77,  and  whose  difference,  multiplied  by  the  lesser,  is  equal  to  121 

\  ms.  i  and  7. 

(17)  To  find  a  number  such  that,  if  you  subtract  it  from  10,  and  multiply  the 
remainder  by  the  number  itself,  the  product  shall  be  21. 

Ans.  7,  or  3» 


QUADRATIC  EQUATIONS.  237 

(18/  To  divide  100  into  two  such  parts  that  the  sum  of  their  square  roots 
may  be  14. 

Ans.  64  and  36. 

(19)  It  is  required  to  divide  the  number  24  into  two  such  parts  that  their 
product  may  be  equal  to  35  times  their  difference. 

Ans.  10  and  14. 

(20)  Tbe  sum  of  two  numbers  is  8,  and  the  sum  of  their  cubes  is  152. 
"What  are  the  numbers  ? 

Ans.  3  and  5. 

(21)  The  sum  of  two  numbers  is  7,  and  the  sum  of  their  4th  powers  is 
641 .     What  are  the  numbers  ? 

Ans.  2  and  5. 

(22)  The  sum  of  two  numbers  is  6,  and  the  sum  of  their  5th  powers  is 
1056.     What  are  the  numbers  ? 

Ans.  2  and  4. 

(23)  Two  partners,  A  and  B,  gained  c£l40  by  trade;  A's  money  was  ? 
months  in  trade,  and  his  gain  was  c€60  less  than  his  stock;  and  B's  money, 
which  was  ^£50  more  than  A's,  was  in  trade  5  months.    What  was  A's  stock  ? 

Ans.  d£l00. 

(24)  To  find  two  numbers  such  that  the  difference  of  their  squares  may 
be  equal  to  a  given  number,  g2 ;  and  when  the  two  numbers  are  multiplied  by 
the  numbers  a  and  b  respectively,  the  difference  of  the  products  may  be  equal 
to  a  given  number,  s2. 

as2 ±  b  Vs4— (a3— 6-)o» 

Ans- ^ — n 

a- — b- 


Js2±a-v/s4— (a3  —  b-)qi 
a2— ft2 

(25)  There  are  two  square  buildings  that  are  paved  with  stones  a  foot 
square  each.  The  side  of  one  building  exceeds  that  of  the  other  by  12  feet, 
and  both  their  pavements  taken  together  contain  2120  stones.  What  are  the 
lengths  of  them  separately  ? 

Ans.  26  and  38  feet. 

(26)  A  and  B  set  out  from  two  towns,  which  were  at  the  distance  of  247 
miles,  and  traveled  the  direct  road  till  they  met.  A  went  9  miles  a  day,  and 
the  number  of  days  at  the  end  of  which  they  met  was  greater  by  3  than  the 
number  of  miles  which  B  went  in  a  day.     How  many  miles  did  each  g6  ? 

Ans.  A  went  117  and  B  130  miles. 

(27)  The  joint  stock  of  two  partners  was  $2080  ;  A's  money  was  in  trade  9 
months,  and  B's  6  months  ;  when  they  shared  stock  and  gain,  A  received 
$1140  and  B  $1260.     What  was  each  man's  stock  ? 

Ans.  $960  and  $1120. 

(28)  A  square  court-yard  has  a  rectangular  gravel  walk  round  it.  The  side 
of  the  court  wants  2  yards  of  being  6  times  the  breadth  of  the  gravel  walk, 
and  the  number  of  square  yards  in  the  walk  exceeds  the  number  of  yards  in 
the  periphery  of  the  court  by  164.     Required  the  area  of  the  court. 

Ans.  256. 

(29)  During  the  time  that  the  shadow  on  a  sun-dial,  which  shows  trae 
time,  moves  from  1  o'clock  to  5,  a  clock,  which  is  too  fast  a  certain  number    f 


238  ALGEBRA. 

hours  and  minutes,  strikes  a  number  of  strokes  equal  to  that  number  of  'ooura 
and  minutes ;  and  it  is  observed  that  the  number  of  minutes  is  less  by  41  than 
the  square  of  the  number  which  the  clock  strikes  at  the  last  time  of  striking 
The  clock  does  not  strike  twelve  during  the  time.     How  much  is  it  too  fast? 

Ans.  3  hours  and  23  minutes. 

(30)  A  and  B  engage  to  reap  a  field  for  d£4  10s. ;  and  as  A  alone  could  reap 
it  in  9  days,  the}'  promised  to  complete  it  in  5  days.  They  fouud,  however, 
that  they  were  obliged  to  call  in  C,  an  inferior  workman,  to  assist  them  for  the 
last  two  days,  in  consequence  of  which  B  received  3s.  del.  less  than  he  other- 
wise would  have  done.     In  what  time  could  B  or  C  alone  reap  the  field  ? 

Ans.  B  could  reap  it  in  15  days,  C  in  18. 

(31)  The  fore  wheel  of  a  carriage  makes  6  revolutions  more  than  the  hind 
wheel  in  going  120  yards  ;  but  if  the  periphery  of  each  wheel  be  increased  1 
yard,  it  will  make  only  4  revolutions  more  than  the  hind  wheel  in  the  same 
space.     Required  the  circumference  of  each.  Ans.  4  and  5. 

(32)  The  intensity  of  two  lights,  A  and  B,  is  as  7  :  17,  and  their  distance 
apart  132  feet.     "Whereabouts  between  is  the  point  of  equal  illumination? 

Ans.  51.595  feet  from  A. 

(33)  The  loudness  of  a  church  bell  is  three  times  that  of  another.  Now, 
supposing  the  strength  of  sound  to  be  inversely  as  the  square  of  the  distance, 
at  what  place  between  the  two  will  the  bells  be  equally  well  heard  ? 

Ans.  .3662  of  distance  between  the  bells  from  the  second. 

(34)  Supposing  the  mass  of  the  earth  to  be  1  and  that  of  the  moon  0.017, 
their  distance  240  thousand  miles,  and  the  force  of  attraction  equal  to  the  mass 
divided  by  the  square  of  the  distance  ;  at  what  point  between  will  a  body  be 
held  in  suspense,  attracted  toward  neither? 

Ans.  27G82.8  miles  from  the  moon. 

(35)  The  hold  of  a  vessel  partly  full  of  water  (which  is  uniformly  increased 
by  a  leak)  is  furnished  with  two  pumps,  worked  by  A  and  B,  of  whom  A  takes 
three  strokes  to  two  of  B's;  but  four  of  B's  throw  out  as  much  water  as  five 
of  A's.  Now  B  works  for  the  time  in  which  A  alone  would  have  emptied  the 
hold;  A  then  pumps  out  the  remainder,  and  the  hold  is  cleared  in  13  hours 
and  20  minutes.  Had  they  worked  together,  the  hold  would  have  been  emp- 
tied in  3  hours  and  45  minutes,  and  A  would  have  pumped  out  100  gallons 
more  than  he  did.  Required  the  quantity  of  water  in  the  hold  at  Bret,  and 
the  hourly  influx  of  the  leak. 

Ans.  1200  gallons  in  the  hold,  120  gallons  of  leakage  per  hour. 

(36)  To  divide  two  numbers,  a  and  b,  each  into  two  parts,  such  that  the  prod- 
uct of  one  part  of  a  by  one  part  of  b  may  be  equal  to  a  given  cumber,  /',  and  the 
product  of  the  remaining  parts  of  a  and  b  equal  to  another  given  number,  j  '. 

Ai  s      ^gb-(p'-p)±  V  \ab-(p'-p)\*-4abp 

26 


ab  +  {p'— p)^  V  \ab  —  {p'—p)\*—4abp 
+  26 


_ gb—{p'—p)Az  V\ab  —  (p'—p)\1—4 abp 
y  2a 


ah+(p—  p)zf  yf\ gb  —  {p'—p)\*—4„br 
2a 


QUADRATIC  EQUATIONS.  239 

(37)  To  find  a  number  such  that  its  square  may  be  to  the  product  of  the 
differences  of  that  number,  and  two  other  given  numbers,  a  and  b,  in  the 
given  ratio,  p  :  q. 


{a-\-b)p±  v'(« — l')2j)2-\-'\abpq 

Ans. — - . 

2{p  —  q) 

(38)  There  is  a  number  consisting  of  two  digits,  which,  when  divided  by 
the  sum  of  its  digits,  gives  a  quotient  greater  by  2  than  the  first  digit ;  but  if 
the  digits  be  inverted,  and  the  resulting  number  be  divided  by  a  number  greater 
by  unity  than  the  sum  of  the  digits,  the  quotient  shall  be  greater  by  2  than  the 
former  quotient.     What  is  tho  number? 

Ans.  24. 

(39)  A  regiment  of  foot  receives  orders  to  send  216  men  on  garrison  duty, 
each  company  sending  the  same  number  of  men  ;  but  before  the  detachment 
marched,  three  of  the  companies  were  sent  on  another  service,  and  it  was  then 
found  that  each  company  that  remained  would  have  to  send  12  men  additional 
in  order  to  make  up  the  complement,  21G.  How  many  companies  were  in  the 
regiment,  and  what  number  of  men  did  each  of  the  remaining  companies  send 
on  garrison  duty  ? 

Ans.  There  were  9  companies,  and  each  of  the  remaining  6  sent  3G  men. 

DECOMPOSITION   OF   THE    TRINOMIAL  X2-\-pX —  q  INTO   TWO    FACTORS    OF    THE 

FIRST   DEGREE. 

193.  If  we  add  to  this  trinomial,  in  order  to  complete  the  square  of  the  first 
two  terms,  the  term  ^p2,  and  afterward  subtract  the  same,  so  as  not  to  change 
the  quantity,  it  becomes 

2-2  +px  -\- 1  p- — I p- — q, 
which  may  be  written  thus  : 

(*+iiOS-Ui>8+?) (2) 

But  the  difference  of  the  squares  of  two  quantities  being  equal  to  the  prod 
uct  of  their  sum  and  difference,  the  expression  (2)  is  equal  to  the  following  • 

(*+&+  VW+q)(x+hp-  V^+q)  •  •  •  (3) 
We  perceive  from  this  expression  that  the  two  factors  of  the  first  degree, 

which  compose  the  trinomial  of  the  second  degree,  are  x  minus  each  of  the 

roots  of  the  equation  of  the  second  degree,  formed  by  putting  this  trinomial 

equal  to  zero. 

Moreover,  by  equating  (3)  to  zero,  we  perceive  that  the  only  way  of  satis 

fying  the  resulting  equation  is  by  making  one  or  other  of  the  factors  of  the 

first  degree,  of  which  it  is  composed,  equal  to  zero. 
The  first, 

X+UJ+  V!^+?=o,  gives  x=—  \p—  Vlp2+q; 

and  the  second, 


x+h7—  VkP2+1=0^  §ives x=—h>+  V\p2+q- 
Hence  there  are  but  two  values  of  x  which  will  satisfy  the  general  equatioa 

x2-\-px — 9=0. 

EXAMPLES. 

1°.  Decompose  the  trinomial  x2— 7z-}-10  into  two  factors  of  the  first  de- 
gree. 


240  ALGEBRA. 

From  the   equation  x* — 7x+10=0  we  find  the  roots  x=5  and  1=2. 
Henco 

a?— 7x+10  =  (x— 5)(x— 2). 
2°.  3x°— 5x— 2. 

Equating  this  trinomial  to  zero,  after  dividing  by  3,  wo  obtain  the  equation 
x2 — §x — §=0,  the  roots  of  which  being  x=2  and  x=  —  |,  we  have 
3a«— 5x— 2=3(x— 2)(x+|)=(x— 2)(3x+l). 

3°.  .7-+5T+3.  Ans.  (*+$—£  V^)(*+5+Wl3). 

4°.  4a*— 4ar+l.  Ans.~(2x— 1)-.« 

5°.  x-— 5x+7.  Ans.  (x— f)*+|. 

194.  To  complete  the  analysis  of  the  2°  degree,  it  would  be  necessaiy  to 
consider  the  case  where  the  unknown  quantities  exceed  the  equations  in  num- 
ber. The  moro  simple  is  that  when  there  is  but  one  equation  and  two  un- 
known quantities.  If  it  be  resolved  with  respect  to  one  of  the  unknown  quan- 
tities, y,  for  example,  an  expression  is  found  generally  containing  x  under  a 
radical ;  so  that,  by  giving  to  x  any  rational  values  whatever,  irrational  values 
would  be  found  for  y.  It  might  be  proposed  to  find  rational  values  for  x,  for 
which  the  corresponding  one  of  y  should  be  rational  also.  But  the  difficulty 
of  this  problem,  unless  it  be  restricted  to  some  very  simple  cases,  is  beyond 
mere  elements.  We  add  one  or  two  here.  For  further  information  upon 
the  subject,  the  student  is  referred  to  the  Theory  of  Numbers,  by  Legendre, 
a  separate  and  veiy  elegant  treatise,  in  one  quarto  volume. 

INDETERMINATE    ANALYSIS    OF    TIIE    SECOND    DEGREE. 

Resolution  in  whole  numbers  of  an  equation  of  Oie  second  degree,  with  two 
unknown  quantities,  which  contains  but  the  first  poicer  of  one  of  the  unknowns. 

195.  The  questions  of  indeterminate  analysis,  which  depend  upon  equations 
of  a  degree  superior  to  the  first,  go  beyond  the  limits  which  we  have  imposed 
on  ourselves  in  the  present  work  ;  but  when  an  equation  of  the  second  degree 
contains  the  second  power  of  but  one  of  the  unknown  quantities,  the  solutions 
of  this  equation  in  whole  numbers  may  be  regarded  as  a  question  of  indeter- 
minate analysis  of  the  first  degree. 

Equations  of  the  second  degree  in  two  unknown  quantities,  which  do  not 

contain  the  second  power  of  one  of  these,  aro  represented  by  the  equation 

mxy+nx*+px+qy=:r (1) 

Resolving  this  equation  with  respect  to  y,  we  find 

—  nx- — px-\-r 

mx-\-q 

We  deduce  from  it,  by  performing  the  division, 

n       nq — mp     m*r-\-mpq — nq* 

J  m     '        ra2      "*"      m2(jnx-{-  q)     ' 

which  gives 

N 
m-y=—  mnx+nq— mp  +  ^^-j (3) 

outting  to  abridge  m^r+mpq — rt^ssN. 


y= — ^ttt, — (2) 


N_ 
should  bo  a  wholo  number;  wo  must,  therefore,  calculate  all  the  divisors  of 


In  order  that  X  and  y  should  bo  wholo  numbers,  it  is  necessary  that  — - 


*  This  presents  a  case  of  what  aro  oaUed  kjuuI  roots 


For  the  first  equation 


INDETERMINATE  ANALYSIS  OF  THE  SECOND  DEGREE.  241 

the  Dumber  N,  and  put  mx-\-q  equal  to  each  of  these  divisors  successively, 
taken  with  the  sign  -f-  and  with  the  sign  — .  If  the  equations  thus  obtained 
furnish  for  x  a  certain  number  of  entire  values,  these  values  are  to  be  substi- 
tuted in  equation  (3);  and  it  is  necessary,  moreover,  in  order  that  y  may  be  o 
whole  number,  that  the  second  member  which  becomes  a  known  quantity 
should  je  divisible  by  m". 

It  is  evident  that  the  number  of  entire  solutions  will  be  very  limited,  and 
that  there  may  not  be  even  one. 

If  this  method  be  applied  to  each  of  the  following  equations, 

2xy— 3r2+  y=l 
5xy=2x  +3?/+ 18 
xy-\-  x°~=2x-\-3y-\-29, 
considering  only  the  positive  solutions,  we  find 

(  .r=0,  y  =  l 
'  \  .r=3,  i/=4. 
( x=l,  y=\0 
For  the  second  equation   ..._.<  x=3,  y=2 

(  x=7,  ?/  =  l. 

c  x=4,  y=21 
For  the  third  equation <     ,-      „ 

If' the  remainder,  after  the  hvision  of  —nx-—px-\-r  by  mx-\-q,  should  be 
zero,  equation  (1)  would  be  of  the  form  (mx-\-q)(ax+by-{-c)  =  0  ;  ai;d  we 
should  have  all  the  solutions  of  this  equation  by  resolving  separately  the  two 
equations  mx-\-q=0,  ax-\-by-\-c  =  0. 

The  method  which  has  just  been  explained  is  applicable  only  in  case  in  is 
not  zero. 

Let  ??i  =  0  ;  equation  (1)  give,s 
nx*+px—r 

y=—-q — (4> 

Suppose  that  one  value  of  x=a  (a  being  a  whole  number)  gives  an  entire 
value  for  y.  If  we  place  x=a-\-qt,  t  being  any  entire  number  whatever,  we 
find 

y=  —  — -I^ (2nat-\-nqP  +pt); 

by  hypothesis,  na?-{-pa—r  is  divisible  by  q ;  the  value  of  y,  corresponding  to 
1=0+ qt,  will  be  then  a  whole  number.  As  this  conclusion  is  true,  what- 
ever be  the  sign  of  t,  it  follows  that,  if  the  equation  admits  of  entire  solutions, 
thoy  will  be  found  to  be  such  as  answer  to  a  value  of  x  between  0  and  q. 
Consequently,  to  obtain  all  the  solutions  in  whole  numbers,  it  will  be  suffi- 
cient to  substitute  for  x  in  the  equation  the  numbers  0,  1,  2,  3,  . . .  q— 1, 
and  each  solution  in  whole  numbers  corresponding  to  one  of  these  numbers 
will  furnish  an  infinite  number  of  others. 

Equation  (4),  in  which  the  object  is  to  find  values  of  x  which  render  the 
polynomial  ?ix--\-px — r  a  multiple  of  the  given  number  q,  M.  Gauss  calls  con- 
gruence of  the  second  degree  ;  so,  also,  the  equation  ax-{-by=c,  in  which  we 
eeek  to  render  ax — c  a  multiple  of  b,  is  a  congruence  of  the  firs'-  degree. 

Further  matter  on  the  subject  of  indeterminate  analysis  will  be  given  in  con- 
nection with  the  theory  of  numbers,  for  which  see  a  subsequent  part  of  the 
work. 

Q 


242  ALGEBRA. 

MAXIMA  AND  MINIMA. 

196.  When  a  quantity  which  is  capable  of  changing  its  valut  attains  such  a 
value  that,  after  having  been  increasing,  it  begins  to  decrease,  cr,  Inning  been 
decreasing,  it  begins  to  increase,  in  the  first  case  it  is  called  a  maximum,  and  in 
the  second  a  minimum.     The  same  quantity  may  have  several  maximum  or 

minimum  values. 

EXAMPLE. 

To  find  what  value  of  x  will  render  the  fraction a  maximum  or 

*2x '2 

minimum. 

Equating  the  given  function  of  x  to  2,  we  have 

t' 0r_i_O  

~-±±Z=z  .-.  x=z+l±  V~:-l- 

"We  perceive  at  once  that  by  making  r  ==  -j-  1  we  have  x=2,  and  that  the 
values  of  2,  a  little  less  than  1,  render  x  imaginary  ;  hence  the  given  expression 
has  a  minimum  value  1  corresponding  to  x=2. 

In  a  similar  manner,  making  z= — 1,  we  have  .7=0;  and  a  negative  value 
of  2,  a  little  smaller  than  1,  would  render  x  imaginary.  But  in  algebra,  nega- 
tive quantities,  which,  without,  regard  to  the  sign,  go  on  increasing,  ought  to  be 
regarded,  when  the  sign  is  prefixed,  as  decreasing;  we  may,  therefore, 
that  a  value  of  2,  a  little  greater  than  — 1,  renders  x  imaginary,  then  z  =  —  1  is 
a  maximum  corresponding  to  x=0. 

As  the  subject  of  maxima  and  minima  is  generally  treated  by  the  aid  of  the 
differential  calculus,  we  shall  not  dwell  further  upon  it  here,  though  it  furnishes 
one  of  the  applications  of  equations  of  the  second  degree. 

THE  MODULUS  OF  IMAGINARY  QUANTITIES. 

197.  We  have  seen  (191)  in  the  equation  of  the  second  degree 

x*+px+q=0, 

that  when  q  is  positive,  and  greater  than  "-— ,  the  roots  are  imaginary.     Replace 

\p  by  — a,  to  avoid  fractions  ;  and  to  express  that  7>-r.  put  g=a54-&'; ;  the 
equation  will  become 

and,  by  the  formula  for  the  solution  of  equations  of  the  second  degree, 

x=aAz  V— &*> 
or 

x=a±bV~^l (I) 

The  absolute  value  of  the  square  root  of  the  positive  quantity  a:-{-b:  is  call- 
ed the  modulus  of  the  imaginary  expression  (1).  For  example,  the  modulus 
of  3— 4  V  — 1  would  be  V^  +  IG,  or  5. 


Two  quantities,  such  as  a-\-bi/ —  1  and  a  —  b  \/  — 1,  which  differ  from  one 
another  only  in   the  sign  of  the   imaginary  part,  are  called  I  mjugatU  of  each 

other.    Two  conjugate  quantities  bars  then  ihe  same  modi  'us. 

If  we   make   //=<),   the    expression  a-{-h  y/  —  1    reduces   to  '/.      Thus,   the 

formula  x =a-\-b  y/  —  1  may  represent  all  quantities  real  or  imaginary,  fl  rep- 
resenting the  algebraic  sum  of  the  real  quantities,  and  b  that  of  the  roellicients 


THE  MODULUS  OF  IMAGINARY  CJANTITIES.  L'43 

of  *J  — l  in  the  imaginary  terms.  When  the  quantity  is  real,  it  has  for  con- 
jugate an  equal  quantity,  and  the  modulus  is  nothing  else  than  the  quantity 
itself,  abstraction  being  made  of  the  sign. 

Now  I  shall  proceed  to  establish  two  propositions  relating  to  moduli,  which 
may  be  often  useful. 

Proposition  I. — The  sum  and  difference  of  any  two  quantities  whatever 
have  a  modulus  comprehended  hetween  the  sum  and  the  difference  of  their 
moduli. 

Let  there  be  two  expressions  a-\-bj  —  1,  a'-\-h'  V  —  1.  Calling  r  and  r' 
their  moduli,  we  have  r2=a2-|-Z;2,  r'2=a"2-\-b"2.  Naming  R  the  modulus  of 
their  sum,  we  have  evidently 

R>=(«+a'y-\-(b+hy 

—  a'iJra'-+lr-{-b'2+2(aa'+bb') 

But  multiplying  r2  by  r'\  we  have 

r2r'2=a2a'2  +  lrb'2+a*b'2+a'2b'i 
=  (aa'  +  bb')-+(ab'—ba'y; 
then  the  numerical  value  of  aa'-\-bb'  is  less  than,  or  at  most  equal  to,  rr'.     Con 
sequently,  it  is  clear  that  R2  is  comprehended  between  the  two  quantities 
ri-^-r'--\-2rr'  and  r°--\-r'~ — 2rr',  or,  what  is  the  same  thing,  between  (r-\-r')9 
and  (r — r')2.     Then  the  modulus  R  is  comprehended  between  the  sum  and 
the  difference  of  the  moduli  r  and  r'. 

The  demonstration  is  precisely  the  same  where,  instead  of  the  sum  of  the 
imaginary  expressions,  we  consider  their  difference. 

Proposition  II. —  The  product  of  two  quantities  has  for  modulus  the  product 
of  the  moduli  of  these  quantities. 

In  fact,  multiplication  gives 

[a  +  b  vr^l)(a'  +  ?/  J~—i)=aa'—bb'-\-{ab'-lrba')  /— i  ; 
and  if  wo  take  the  modulus  of  this  product,  we  find,  conformably  to  the  enun- 
ciation, 


■J{aa'  —  bb'y+{ab'-\-ba')i=y/aia'-+b-b'2-\-a2b'i-{-b2a'i 

=  V(a>+b2)(a'*  +  b'2). 

Corollary. — Then  the  product  of  any  number  of  factors  whatever  must 
nave  for  modulus  the  product  q  of  the  moduli  of  all  the  factors.  Then  the 
*>lh  power  of  an  imaginary  expression  has  for  modulus  the  nth  power  of  the 
modulus  of  that  expression. 

The  aoove  nomenclature  and  propositions  are  from  Cauchy,  who  exhibits  in 
a  remarknble  manner  the  efficiency  of  imaginary  expressions  as  instruments  in 
the  investigation  of  the  properties  of  real  quantities.  The  following  is  a 
specimen  : 

If  two  numbers,  of  which  each  is  the  sum  of  two  squares,  be  multiplied  to- 
gether, the  product  must  also  be  the  sum  of  two  squares. 

Let  the  two  numbers  be 

a2+62anda'2-|-Z>'2. 
The  first  of  these  may  be  considered  as  the  product  of  the  factors 

fl-f-6  /  — 1  and  a  —  b  y/  — I, 
and  the  second  as  the  product  of  the  factors, 

a'+b'  -/^T  and  a'—b'  /^l  ; 


241  ALGEBRA. 

so  that  the  product  of  the  proposed  numbers  will  be  the  product  of  the  fout 

fuctors  

a  +  b  V^-T,  a  —  b  ^"^1,  a'+b'  yf  —  1,  a'  —  b'  y/ —  1. 
Actually  multiplying  the  first  and  third,  and  then  the  second  and  fourth,  we 
have  the  following  pair  of  conjugate  expressions,  viz., 

(aa'  —  bb')  +  (ab'+ba')  yf^l%  {aa' —  bb')  —  (ab' +ba')  y/^-i, 
of  which  the  product  is 

{aa'  —  bb'Y+{ab' +ba'y, 
which  is,  therefore,  the  product  of  the  original  numbers,  and  proves  that  that 
product  must,  like  each  of  the  proposed  factors,  be  the  .sum  of  two  squares. 

If  we  interchange  the  numbers  a  and  b,  or  the  numbers  a',  6',  the  terms  of 
the  product  just  deduced  will  be  different;  thus,  putting  a'  for  b',  and  b'  for 
a',  which  produces  no  essential  change  in  the  proposed  numbers,  we  have 
(a-  +  b:){a'-2+b'i)  =  {aa'  —  bb')-+(ab'+ba')-  =  {ab'  —  ba'y-  +  (aa'  +  bb')-. 
Consequently  there  are  two  ways  of  expressing  by  the  sum  of  two  squares 
the  products  of  two  numbers,  each  of  which  is  itself  the  sum  of  two  squares ; 
thus, 

(52+2-)(33+2-)  =  ll-+16-  =  4-+193 
(2s-r.l2)(324-2-)=  4--f   7:  =  1-+   83 
&c,  &c. 

METHOD   PROPOSED  BY   MOUREY   FOR  AVOIDING    IMAGINARY    QUANTITIES.* 

198.  Objections  have  been  made  to  results  obtained  by  the  calculus  of  imag- 
inary expressions.  The  rules  observed  in  the  calculus,  it  is  said,  have  only 
been  demonstrated  for  real  magnitudes;  it  is  by  mere  analogy  that  they  are  ex- 
tended to  the  case  of  imaginary  quantities ;  we  may,  therefore,  raise  reasonable 
doubts  as  to  the  exactitude  of  the  results  thus  deduced. 

M.  Mourey,  who  has  been  much  occupied  with  these  difficulties,  has  sought 
to  free  analysis  from  them  entirely,  in  a  work  published  in  1828,  entitled  the 
True  Theory  of  Negative  Quantities  and  of  the  so-called  Imaginary  Quanti- 
ties. Without  entering  into  long  details,  we  shall  endeavor  here  to  give  an 
idea  of  the  methods  proposed  by  this  author. 

Let  us  resume  the  expression  a-]-b.  -y/ — 1<  and  give  it,  at  first,  the  form 

b 


^^{t^+t^^-1] 


If  wo  take  the  sum  of  the  squares  of  tin1  fractions,  which  are  between  the 
brackets,  we  find  that  this  sum  is  equal  to  1  ;  and  from  thence  we  conclude  that 
these  two  fractions  can  be  regarded  as  being  the  .sine  and  eosim  of  a  same 
angle  a.  Designato  also  the  modulus  V"'  +'  by  A  ;  the  imaginary  expres- 
sion can  be  put  under  the  form  A(cos  a-|-  ■/  —  1  sin  a).  Considering  that 
this  expression  contains  really  but  two  quantities,  the  modulus  A  and  the 
angle  «,  M.  Mourey  proposes  to  regard  the  modulus  A  a;  expressing    the 

length  of  a  right  line  O  A.  a  sing 

the  angle    LOX(  which  this  line  D 
with  a  fixed  axis  OX.      In  other  w. 

the  modulus  A  represents  a  lme  of  a  cer- 
tain length,  which  at  first  lay  upon  the 
axis  < »  Vaud  which,  by  making  a  move- 

•  To  luidersturul  this,  a  knowledge  of  the  first  principles  of  Trigonometry  is  accessary 


MOUREY'S  METHOD  FOR  AVOIDING  IMAGINARY  QUANTITIES.    245 

merit  round  the  origin  O  upward,  has  departed  from  this  axis  by  an  angle  a. 
M.  Mourey  gives  the  name  verser  to  this  angle,  or,  rather,  to  the  arc  which 
measures  it;  and  then,  instead  of  the  imaginary  expression,  he  writes  simply 
Aa,  a  uotation  very  suitable  to  recall  at  the  same  time  the  modulus  A  and  the 
verser  a.  He  proposes  even  to  give  the  name  route,  or  way,  to  the  length  O  A, 
placed  in  its  true  position  with  regard  to  OX,  so  that  A  verser  a,  orAa,ia  the 
route  from  O  toward  A. 

As  a  line  can  make  around  the  origin  O  as  many  revolutions  as  we  please, 
and  that,  also,  as  well  by  commencing  its  rotation  below  as  well  as  above  O  X, 
it  follows  that  the  verser  may  pass  through  all  states  of  magnitude,  and  be  aa 
well  negative  as  positive.  It  will  bo  positive  when  the  movement  of  the  line 
shall  have  commenced  above  ;  it  will  be  negative  when  the  movement  com- 
menced below.  From  this  it  follows  that  the  same  route  can  be  represented 
with  a  verser  which  is  positive,  or  one  which  is  negative,  provided  that  the 
sum  of  tho  versers,  abstraction  being  made  of  the  signs,  is  360°. 

From  the  preceding  conventions  it  results  that  a  way  can  be  represented  by 
giving  to  tho  length  A  an  infinity  of  different  versers.  Suppose,  to  fix  the 
ideas,  that  O  A  should  be  a  determinate  way,  and  that  then  the  verser  A  OX 
should  be  an  acute  angle  a  ;  it  is  evident  that  the  position  of  O  A  will  undergo 
no  change  if  we  add  or  subtract  from  a  any  number  whatever  of  entire  cir- 
cumferences. Thus  is  established  this  important  remark,  that  if  we  desig- 
nate by  2tt  an  entire  circumference,  or  360°,  and  by  n  any  whole  number 
whatever,  positive  or  negative,  the  expression  A27r«-j-°  will  represent  tho 
same  route  as  Aa ;  this  is  expressed  by  the  equality 

When  we  give  to  A  a  verser  equal  to  zero,  the  length  A  lies  upon  the  lino 
OX.     When  the  verser  is  equal  to  n  or  180°,  this  length  is  found  in  the  op 
posite  direction,  O  X' ;  then  it  is  nothing  else  than  the  negative  quantity  — A. 
Thus  we  ought  to  regard  as  altogether  equivalent  the  two  expressions  — A 
and  Air. 

After  these  preliminaries,  M.  Mourey  establishes  the  rules  of  algebraic 
calculus  ;  then  he  passes  to  equations,  and  reconstructs  algebra  thus' entirely. 
I  shall  not  follow  this  author  in  all  his  details  ;  I  shall  confine  myself  to  the 
developments  necessary  to  explain  here  what  sense  the  new  algebra  attaches 
to  the  old  imaginary  expression  -/ — A2.  I  shall  seek,  first,  the  rule  to  be 
followed  in  the  multiplication  of  any  two  quantities  whatever,  Aa  and  B/?. 
Here  the  two  factors  are  the  magnitudes  A  and  B,  measured  upon  two  lines 

O  A  and  O  B.  which  make,  with  a  fixed  axis 
OX,  angles  A  OX,  BOX,  represented  by  tho 
'A'  versers  a  and  (8.     It  is  necessary,  then,  first 
of  all,  to  give  to  the  definition  of  multiplica- 
tion the  extension  suitable  to  render  it  appli- 
cable to  the  case  in  question.     But,  consider- 
ing that,  the  multiplier  B/3  indicates  a  line  B, 
which  departs  from  the  fixed  line  O  X  by  an 
angle  equal  to  /?,  M.  Mourey  regards  multi- 
plication sis  having  for  its  object  to  take  at 
first  the  length  A  in  its  actual  direction  as  many  times  as  there  are  units  in  B, 
and  to  turn  the  new  line  O  A'  around  the  point  O,  to  depart  from  this  direc- 


246  ALGEBRA. 

Hon  by  an  angle  enual  to  ft,  and  to  give  it  the  position  O  0  .     From  this  it  fol- 
lows that,  in  designating  by  AB  the  product  of  the  two    nugnitudes,  obstrac- 
tion  being  made  of  all  idea  of  position,  the  product  sougl  t  will  bo  (AB)a-|-' 
Thus  we  have 

A«XB/3=(AB)rt-f/3; 
that  is  to  say,  wo  multiply  the  moduli  according  to  the  ordinary  rules  of  a 
metic,  and  tale  the  sum  of  the  verscrs. 

If  the  two  versers  are  equal  to  it  or  130°,  we  shall  have  \-  X  B-=(AB).'-. 
But  A^  and  Bw  are  nothing  else  than  — A  and  — B,  and  (AB)2t  is  the  same 
tiling  as  -|-AB  ;  then  — Ax  — B  =  -|-AB.  This  is  the  known  rule,  —  by  — 
gives  -f- . 

According  to  this  rule,  the  square  of  Aa  will  be  (A-)2a  ;  that  is  to  say, 
take  the  square  of  the  modulus  and  double  the  verser.     Then,  reciprocally,  the 
square  root  is  obtained  by  extracting  the  square  root  of  the  modulus  without  re- 
garding the  verser  ;  then  take  half  the  verser. 

Let  us  come  now  to  the  interpretation  of  the  imaginary  expression  ■/  —  A-\ 
For  this  purpose,  let  us  observe,  first,  that  it  is  equivalent  to  y/ (A:z)2n~-\-- ; 
then  extracting  the  square  root, 

V  —  A-  =  An --f- '  T- 
If  n  is  even,  the  verser  nT-k-'."  places  the  length  A  in  the  same  position  as 
\n ;  that  is  to  say,  in  the  position  OP,  perpendicular  ;    <  >  \. 
If  n  is  uneven,  the  verser  nrr-\-\-  will  place  the  length  A  in 
a  position  O  P',  perpendicular  to  O  X,  but  bebw.     Thus,  in 


O      X  the  system  of  M.  Mourey,  the  expression  -/  —  A*  offers  no 
longer  to  the  mind  any  idea  of  impossibility.     It  represeuta 
P'  two  routes,  OP  and  OP',  equal  and  opposite,  both  perpen- 

dicular to  the  fixed  axis  O  X. 


PERMUTATIONS  AND  COMBINATIONS. 

199.  The  Permutations  of  any  number  of  quantities  are  tin-  changes  which 
these  quantities  may  undergo  with  respect  to  their  order. 

Thus,  if  we  take  tlie  quantities  </,  /',  c;  then  </!>>•,  acb,  bar,  bra.  cab,  ct,a 
are  the  permutations  of  these  three  quantities  taken  all  together;  ab,  ac,  ba, 
be,  ca,  cb  nre  the  permutations  of  those  quantities  taken  two  and  two;  a,  b,  i 
are  the  permutations  of  these  quantities  taken  singly,  or  <>n,  and  one,  &cc. 

The  problem  which  we  propose  to  resolve  is,  . 

200.  To  find,  the  number  of  the  permutations  of  n  quantities,  taken  p  ani/  p 
together. 

Let  a,  6,  c,  d, k,  bo  the  n  quanti'  es. 

The  number  of  the  permutations  of  these  /;  quantities  taken  singly,  or  one 
and  one,  is  manifestly  n. 

The  Dumber  of  the  permutations  <>t  these  n  quantities,  taken  two  and  two 

together,  will  he  n(n  —  1).      For,  since  there  are  //  quantities, 

a,  /',  <.</, k. 

If  we  remove  a  there  will  remain  (n  —  1)  quantities, 

b,  c,  d, k. 


PERMUTATIONS  AND  COMBINATIONS.  247 

Writing  a  before  each  of  these  (n  —  1)  quantities,  we  shall  have 

ab,  ac,  ad, ak ; 

that  is,  (n  —  1)  permutations  of  the  n  quantities  taken  two  and  two,  in  which  a 
stands  first.  Reasoning  in  the  same  manner  for  b,  we  shall  have  (n—  1)  per- 
mutations of  the  n  quantities  taken  two  and  two,  in  which  b  stands  first,  and 
so  on  for  each  of  the  n  quantities  in  succession;  hence  the  whole  number  of 
permutations  will  be 

n(n —  1). 

The  number  of  the  permutations  of  n  quantities,  taken  three  and  three  to- 
gether, is  n(n — \)(n — 2).  For  since  there  are  n  quantities,  if  we  remove  a 
there  wil.  remain  (?» —  1)  quantities;  but,  by  the  last  case,  writing  (n —  1)  for 
n,  the  number  of  the  permutations  of  (/<  —  1)  quantities,  taken  two  and  two,  is 
(n  —  l)\n — 2);  writing  a  before  each  of  these  (n  —  l)(n — 2)  permutations, 
we  shall  have  («  —  l)(n — 2)  permutations  of  the  n  quantities,  taken  three  and 
three,  in  which  a  stands  first.  Reasoning  in  the  same  manner  for  b,  we  shall 
have  (n  —  l)(n — 2); permutations  of  the  n  quantities,  taken  three  and  three,  in 
which  b  stands  fivst,  and  so  on  for  each  of  the  n  quantities  in  succession  ;  hence 
the  whole  number  of  permutations  will  be 

n{n  —  l){n—  2). 

In  likq,  manner,  wo  can  prove  that  the  number  of  permutations  of  n  quan- 
tities, ta.,en  four  and.  four,  will  be 

n(n  —  l){n— 2)(n— 3). 

Upon  examining  the  above  results,  we  readily  perceive  that  a  certain  rela- 
tion exists  between  the  numerical  part  of  tho  expressions  and  the  class  of  per- 
mutations to  which  they  correspond. 

Thus  the  number  of  permutations  of  n  quantities,  taken  two  and  two,  is 

n(n —  I),  which  may  be  written  under  the  form  n(n  —  2-j-l). 
Taken  three  and  three,  it  is 

n(7i  —  l)(n — 2),  which  may  be  written  under  the  form  n(n — l)(ra — 3-f-l). 
Taken  four  and  four,  it  is 
n(n  — l)(ra — 2)(n — 3),  which  may  be  written  under  the  form  n(n — \){n — 2) 

(n-4  +  1). 
Hence,  from  analogy,  we  may  conclude  that  the  number  of  permutations 
of  n  things,  taken  p  and  p  together,  will  be 

n(n  — 1)(»  —  2)(n  —  3)  . (,i_p_{-l). 

In  order  to  demonstrate  this,  we  shall  employ  the  same  species  of  proof 
already  exemplified  in  (Arts.  23  and  78),  and  show  that,  if  the  above  law  be 
assumed  to  hold  good  for  any  one  class  of  permutations,  it  must  necessarily 
hold  good  for  the  class  next  superior. 

Let  us  suppose,  then,  that  the  expression  for  the  number  of  the  permuta- 
tions of  n  quantities,  taken  {p  —  1)  and  (p  —  1)  together,  is 

n(n  —  l){n—2){n—3)  .  .  .  \  n  —  (p  —  l)  +  l }  ...  (A) 
It  is  required  to  prove  that  the  expression  for  the  number  of  the  permuta- 
tions of  n  quantities,  taken  p  and  p  together,  will  be 

n(n— l)(n— 2)[n — 3) (n— p+1). 

Remove  a,  one  of  the  n  quantities  a,  b,  c,  d k,  then,  by  the  ex- 
pression (A),  writing  (n  —  1)  for  n,  the  number  of  the  permutations  of  the 
(n_l)  quantities  b,  c,  d k,  taken  (p  —  1)  and  (^  —  1),  will  be 


248  ALGEBRA. 

(»— 1)(»— 2)(n— 3) [(*— ^-(j>— l)+lj, 

or 

(n—l)(w— 2)(n-3) (m— J>+1). 

Writing  a  before  each  of  these  (n — l)(n — 2)(n — 3) ("—/>  +  !) 

permutations,  we  shall  have  (n — l)(n — '-')(«—  3) (n—p+l)  per- 
mutations of  the  n  quantities,  in  which  a  stands  first.  Reasoning  in  the  same 
manner  for  b,  we  shall  have  (n  —  l)(n  —  2)(n— 3) (n — ]>-\-l)  per- 
mutations of  the  n  quantities,  in  which  b  stands  first;  and  so  on  for  each  of 'he 
n  quantities  in  succession  ;  hence  the  whole  number  of  permutations  will  bo 

»(»-l)(n-2)(»-3) (n-p+l) (1) 

Hence  it  appears  that,  if  the  above  law  of  formation  hold  good  for  any  one 
class  of  permutations,  it  must  hold  good  for  the  class  next  superior  ;  hut  it  has 
been  proved  to  hold  good  when  p=2,  or  for  the  permutations  of  ;i  quantities 
taken  two  and  two  ;  hence  it  must  hold  good  when  p  =  3,  or  for  the  permuta- 
tion of  n  quantities  taken  three  and  three  ;  .-.  it  must  hold  good  when  />  =  4, 
and  so  on.     The  law  is,  therefore,  general. 

EXAMPLE. 

Required  the  number  of  the  permutations  of  the  eight  letters  a,  b,  c,  d,  e, 
f,  g,  ft,  taken  5  and  5  together. 

Here  n=8,  p=o,  n — p-\-l=A  ;  hence  the  above  formula 

n{n  —  l)(n  —  2)  ....  (n—  p  +  l)  =  6x  7  X  G  X5  X  4=6720, 
the  number  required. 

201.  In  formula  (1)  let  p=n,  it  will  then  become 

n(n— 1)(»— 2) 2.1, 

or 

1.2.3 (n  — 1)» (2) 

which  expresses  the  number  of  the  permutations  of  n  quantities  taken  all 
together.* 

EXAMPLE. 

Required  tho  number  of  the  permutations  of  the  eight  letters  a,  b,  c,  </,  c, 

fi  £i  >>■ 

Here  n=8;  hence  the  above  formula  (2)  in  this  case  becomes 

1.2.3.4.5.6.7.8=40320, 

the  number  required. 

202.  The  number  of  the  permutations  of  n  quantities,  supposing  them  all 
different  from  each  other,  we  have  found  to  be 

1.2.3 (n— l)n. 

But  if  tho  same  quantity  be  repeated  a  certain  number  of  times,  then  it  is 

manifest  that  a  certain  Dumber  of  the  above  permutations  will  become  identical 

Thus,  it"  one  of  the  quantities  be  repeated  a  times,  the  number  of  identical 

permutations  will  be  represented  by  1.2.3 <i ;  and  hence,  in  order  ti> 

*  Many  writ-  the  term  perm  toth  the  quan- 

tities iiiv  taken  nil  together,  nml  iriye  the  title  ":  tmentt  or  variations  u<:  i 

of  the  n  quantities  when  taken  two  mid  two,  three  and  ■ 

traduction  of  these  nilditional  designations  appears  unnecessary  ;  bat,  in  a  wrord 

perm  ■  .  we  must  always  I  to  mean  those  rc\<<  v  (or- 

muln  (8),  unless  the  contrary  !»■  specified, 


PERMUTATIONS  AND  COMBINATIONS  24 

obtain  the  number  of  permutations  different  from  each  other,  we  must  divide 

(2)  by  1.2.3 a,  and  it  will  then  become 

1^2.3 (n  —  l)n 

1.2.3...^... a        ' 

If  one  .if  the  quantities  be  repeated  a  times,  and  another  of  the  quantities 

be  repeated  /3  times,  then  wo  must  divide  by  1.2 aXl.2 /? ; 

and,  in  general,  if  among  the  n  quantities  there  be  a  of  one  kind,  (1  of  another 
kind,  y  of  another  kind,  and  so  on,  the  expression  for  the  number  of  the  per- 
mutations different  from  each  other  of  these  n  quantities  will  be 

1.2.3 n 

1.2 oXl-2 /3xl.2 y,  &c. *  ' 

KXAMPLK   I. 

Required  the  numbers  of  the  permutations  of  the  letters  in  the  word  algebra. 
Here  n=7,  and  the  letter  a  is  repeated  twice;  hence  formula  (3)  becomes 

1.2. 3. 4. 5. G. 7 

— — • =  2520,  the  number  required. 

1.2  • 

EXAMPLE  II. 

Required   the   number  of  the  permutations  of  the  letters  in  the  word 
caifacarataddarada. 

Hero  ra=18,  a  is  repeated  eight  times,  c  twice,  d  thrice,  r  twice  ;  hence  the 
number  sought  will  be 

1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18 


1.2. 3. 4. 5. 6. 7. 8X1-2X1. 2. 3x1-2 


=6616209G00. 


EXAMPLE   III. 

Required  the  number  of  the  permutations  of  the  product  a*  by  c%  written  at 
full  length. 

Here  n=x-\-y-{-z,  the  letter  a  is  repeated  x  times,  the  letter  b,  y  times, 
and  the  letter  c,  z  time^  ;  the  expression  sought  will,  therefore,  be 

1-2-3 CH-y+z) 

1.2. 3 x  X  1  •  2 .  3 y  X  1  •  2 . 3 z       ' 

203.  The  Combinations*  of  any  number  of  quantities  signify  the  different 
collections  which  may  ba  formed  of  these  quantities,  without  regard  to  the 
order  in  which  they  are  arranged  in  each  collection.  Each  combination  must, 
therefore,  have  one  letter  different  from  any  other  of  the  combinations. 

Thus  the  quantities  a,  b,  c,  when  taken  all  together,  will  form  only  one 
combination,  aba  ;  but  will  form  six  different  permutations,  abc,  acb,  bac,  bca, 
cab^,  cba  ;  taken  two  and  two,  they  will  form  the  three  combinations  etb,  ac,  be, 
and  the  six  permutations  ab,  ba,  ac,  ca,  be,  cb. 

The  problem  which  we  propose  to  resolve  is, 

To  find  tlic  number  of  the  combinations  of  n  quantities,  taken  p  and  p  to- 
gether. 

Each  of  these  combinations  of  p  quantities  being  separately  permutatod,  will 
furnish  1.2. 3... p  permutations,  which,  multiplied  by  the  whole  number  of 
combinations,  will  give  the  whole  number  of  permutations  of  n  quantities,  taken 

*  Where  numerical  or  literal  factors  are  combined,  the  term  combination  may  be  con- 
sidered as  signifying  the  same  as  product. 


250  ALGKBRA. 

p  and  p.     Therefore  tho  latter,   namely,  the  whol„  numbe-  of  permutations, 

or  ii(i)  —  lj(n — 2) (n —  P+1)-  divided  by  the  number  rf  permutations  of 

atch  combination,  or  1  .2 .:;...  p.  will  give  the  number  of  combinations  of  n 
quantities,  taken  p  and  p.     Denoting  it  by  C,  we  have 

n(/i-l)(»-2) (»-p+l) 

c~  r^3 •  •  {i—i)p  "•  ■ (l) 

20  J.  There  is  ;i  Bpecies  of  notation  employed  to  denote  permutations  and 
combinations,  which  is  sometimes  used  with  advantage  from  its  com 
The  number  of  the  permutations  of  n  quantiti        .     a  p  and  p, 

are  repre  ented  by (n^'j) 

The  number  of  the  permutations  of  n  quantities,  taken  all  togeUier, 

are  represented  by ("i'") 

The  number  of  the  combinations  of  n  quantities,  taken  p  and  p, 

are  represented  by ("'/") 

and  so  on.     It  is  manifest  that  the  above  proposition  may  be  expressed  accord- 
ing to  this  notation  by 

InPp) 

(nCp)=K—J-!.. 

M.  Cauchy  employs  the  notation  (m)„  to  express  the  number  of  com 
tions  of  m  letters,  taken  n  at  a  time.     The  German  notation  ame  is 


When  the  scries  of  natural  numbers,  or  the  letters  of  tho  alphabet  up  to 
any  required  number,  are  to  be  permuted  or  combined,  an  abb:  nota- 

tion has  been  employed  as  follows: 

P(l,  2,  3)  stands  for  123,  132,  213,  231,  312,  321. 

P(1..4)  stands  for  12,  13,  14,  21,  23,  24,  31,  32,  34,  41,  42,  43. 

3 

C(a...e)  stands  for  abc,  abd,  abe,  acd,  ace,  ade,  bid,  bec,  bde, 
If  one  or  more  of  the  numbers  or  letters  may  be  repeated,  this  can  also  be 
expressed  in  the  notation.     Thus, 
P(l,  1,  2)  =  112,  121,  211. 

P(l,  1,  2,  3)  =  11,  12,  13,  21,  23,  31,  32. 

C(l,  1,  2,  2,  3)  =  112.  113,  122,  12 

If  all  the  letters,  numbers,  or  single  things  may  be  repeated  an  equal  num- 
ber of  times,  this  can  be  expressed  with  the  aid  of  an  exponent;  thus, 

C(l,  2,  3)\  P(0,  1,-Jr.  C(1..7)». 
205.  If  n  single  things  be  arranged  in  combinations  of  k,  or  ofn — k=r,  \}\» 
number  of  combinations  in  either  case  will  be  die  same,  i.  >-., 

ffc=w(«-l)...(n-fr+l)_/,^w(/>-l)..  .(n-r+1) 
n  1.2.3...*  n  L.2.3..\r 

for  every  new  combination  of  k  letters  must  leave  a  new  one.  ofr  letters* 
By  a  similar  reasoning,  if  n  be  divided  into  three  parts,  the  fust  A-,  tin-  second 

r.  and  the  third  8,  it  may  be  shown  that 

exc  =cxc  =rx(k'  ,  &c 

n  n— t  ii  n     k  ii  i 

20fi  Cases  may  occur  in  which  not  all  p  ssible  combinations,  but  only  such 


PERMUTATIONS  AND  COMBINATIONS.  251 

as  fulfill  certain  conditions,  are  required.  Mnny  such  may  be  imagined.  For 
instance,  where  the  numbers  to  be  combined  increase  by  a  common  difference. 
or  by  a  common  ratio,  as  1357,  2468.  or  124,  or  248.  The  most  useful  case 
is  where  the  number  in  each  combination  must  amount  to  the  same  sum.  The 
method  of  proceeding  in  this  case  is  to  fill  up  all  the  places  except  the  last  with 
the  lowest  numbers,  the  last  place  being  occupied  by  the  supplementary  num- 
ber necessary  to  produce  the  given  sum ;  then  diminishing  the  last  number 
and  increasing  one  of  the  preceding  by  the  same  amount,  taking  care  not  to 
allow  a  lower  ever  to  follow  a  higher  number.     We  give  examples  of  such 

k 

combinations,  the  general  formula  for  which  is  rC(l n). 

(1)  "°C(1...7)=127,  13G,  145,  2:)r>. 

(2)  "C(1...8)=1238,  1247,  1256,  1346,  2345. 

(3)  &C(0..5)«=0005,  0014,  0023,  0113,  0122,  1112. 

(4)  «>C(3....)rc=33338,  33347,  33356,  33446,  33455,  34445,  44444. 

It  is  easy  to  be  perceived  that  in  two  cases  this  kind  of  combination  is  im- 
possible. 1°.  When  the  highest  form  does  not  amount  to  the  required  sum; 
and,  2°.  When  the  lowest  form  exceeds  it,  as  in 

i"C(123)«,  or  ioC(4...)n. 

207.  Similar  conditions  may  be  imposed  upon  permutations.  In  order  that 
the  permutations  of  a  given  series  of  numbers,  taken  a  certain  Dumber  at  a 
time,  should  amount  always  to  a  given  sum,  the  same  rule  will  apply,  with  this 
difference,  that  lower  numbers  may  follow  higher;  in  other  words,  the  com- 
binations formed  by  the  previous  rule  may  each  be  permuted. 

The  following  examples  will  render  this  more  intelligible : 

(1)  oP(1..8)  =  18,  27,  36,45,  54,  63,  72,  81. 

(2)  tP(1...)  =  124,  142,  214,  241,  412,  421. 

(3)  6p(l...)n  =  1113,  1122,  1131,  1212,1221,  1311,  2112,  2121,2211,3111. 

(4)  iP(0..)n=013,  022,  031,  103,  112.  121,  130,  202,  211,  220,  301,  ",10. 

Under  this  head,  also,  two  contradictory  cases  occur:  1°.  When  the  high- 
est form  amounts  to  too  little  ;  and,  2°.  When  the  lowest  form  amounts  ft  too 
much.     As,  for  instance,  in 

9p/1..4)n,  or''P(5...)«. 

208.  The  applications  of  the  theory  of  permutations  and  combinations  are 
numerous.  One  of  the  most  useful  is  the  determination  of  the  coefficients  of 
a  series  of  the  form 

a  +  bx+ c.r2+ rf^-f  ex*-f- . . .  +  kx" . .  .,* 

especially  the  coefficients  of  the  binomial  formula,  the  method  of  determining 
which,  by  the  theory  of  permutations  and  combinations,  will  be  given  here- 
after. 

Another  extensive  application  of  the  theory  of  permutations  and  combina- 


*  These  coefficients  are  supposed  to  depend  upon  some  given  law.  A  common  case  is 
when  the  number  of  {actors  combined  in  each  coefficient  is  indicated  by  the  exponent  of 
the  letter  of  arrangement,  x. 


252  ALGEBRA. 

tions  is  to  be  found  in  geometric  relations,  such  as  where  the  comlinations  of  a 
certain  number  of  points,  lines,  angles,  cVe.,  from  among  a  given  number  of 
these,  are  required. 

Not  less  useful  is  this  theory  in  natural  science  :  as  in  crystalography.  when 
tin-  manifold  tonus  of  crystals  are  required;  in  chemistry,  when  the  various 
combinations  of  chemical  elements;  and  in  music,  of  consonant  tones,  6zc. 

Jim  perhaps  its  most  important  use  is  in  the  doctrine  of  chances,  or,  as  it  is 
mathematically  named,  the 

CALCULUS    OF   PROBABILITY 

The  nut  linos  of  this  extensive  subject  we  shall  here  briefly  indicate,  referring 
the  studenl  for  further  information  to  the  admirable  treatises  of  La  Place 
and  Lacroix,  and  to  the  practical  work  ofDe  Morgan. 

I.  Lei  there  be  among  m  possible  cases  g,  which,  as,  fulfilling  certain  requi- 
sitions, are  considered  as  favorable,  {m — g)  =  u  unfavorable.     Then  the 

of  the  favorable  to  all  possible  cases  is  called  the  mathematical  jnobalii '<!  /  for 
the  occurrence  of  a  favorable  case.  The  ratio  of  the  unfavorable  to  all  possi- 
ble cases  is  the  mathematical  improbability  of  the  occurrence.  If  the  first  be 
expn  s  e  1  by  u;  the  second  by  »,  then 

e;                u 
7/=—  and  v=— (I.) 

The  probability  is,  therefore,  the  less,  the  smaller  the  number  of  the  fa- 
vorable in  comparison  with  that  of  all  possible  cases,  and  vice  versa.     Should 
all  possible  cases  be  favorable,  then  10=1,  which  is,  therefore,  the  expre 
for  certainty.     Thus  the  mathematical  probability  and  improbability  of  a  pic- 
tured card,  of  which  there  are  12,  being  drawn  from  52,  are  expressed  by 

_12_3       _40_10 
,r==52  =  13'  'y=52~13; 
that  of  drawing  one  card  from  52, 

52 
t*  =  -  =  l. 

II.  Let  there  be  among  m  possible  cases  g  favorable,  of  different  (first,  sec- 
ond, third,  cVc.)  kinds,  expressed  by  gl%  g2,  g3,  &c,  the  partial  probabilities 
by  to,,  to3,  to3,  &c.  ;  then 

U-  =  Wl+lV._,-\-H',-\-,\c,= (II.) 

that  is,  the  probability  of  one  of  several  different  kinds  is  equal  to  the  sum  of 

their  partial  probabilities.     Thus,  for  the  probability  of  one  of  tin-  six  faces  of 

a  die,  marked  1,  2,  or  3,  being  thrown,  we  have 

1  1  1 

«,>=-,  «,,=-,«,,=-; 

l     l     l     :;     i 
•••w=6+6+6=6=2' 

III.  Let  the  occurrence  be  favorable  only  on  the  supposition  that  two  or 
more  of  the  single /avordble  cases  concur,  then  the  formula  for  the  compound 
probability  is 

g,  XiS,Xr    ••  .... 

'  -  J  7/1  ,  X  m     X  "l  :i  •  • 

in  which  /«,,  /(<„,  7/i;i,  (Vc,  express  the  poss'blo  cases  of  the  part  nl  occurn n- 


CALCULUS  OF  Pfl,OB  ABILITIES.  253 

ces  ;  that  is,  the  probability  of  the  compounc  occurrence  is  equal  to  the  piod- 
ucts  of  the  partial  probabilities.  For  as  each  of  them,  may  concur  with  each 
of  the  m„  cases,  there  will  be  m,X'«2  possible  cases,  which,  by  the  super- 
vening of  m3  new  cases,  increase  to  m,X»ijX?w3,  and  so  on.  The  same 
reasoning  applies  to  the  favorable  cases  gx,  g2,  g3,  &c,  from  whence,  by  the 
principles  already  established,  results  formula  (III.).  Let  it  be  required,  for 
example,  to  draw  out  of  a  vase  which  contains  the  numbers  1,  2,  3,  4,  5,  and  6, 
first  1*  then  either  2  or  3,  and,  finally,  4,  5,  or  6,  in  three  drawings  ;  the  prob- 
ability is  expressed  by 

1      2     3      1 
™_-x  5X4=05. 

If  the  partial  occurrences  are  equal  (that  is,  repetitions  of  the  same),  then 

«>==  I  —  j  .     Thus,  if  with  each  of  three  dice,  G  shall  be  thrown, 

W=Q  =2l6- 

IV.  Should  there  be  m  possible  cases,  of  which  .g  are  favorable  and  u  un- 
favorable, and  of  these  k-\-r  tire  to  occur,  so  that  k  of  the  favorable,  with  r  of  the 
unfavorable,  must  come  in  juxtaposition,  then  the  expression  for  the  probabili- 
ty of  the  occurrence  of  eveiy  such  order  is 

/jA/g__l\     /g— fc+l\      /    u    \(    u—\    \     I    u— r+1     \ 
w—  W  \^I-[)-\m  —  k+l)  X  \m^k)  WTT-^Tr'Am— k— r+1/  (IV^ 

This  depends  on  (III.),  each  of  the  factors  in  the  above  value  of  w  ex 
pressing  the  partial  probability  of  the  single  occurrence  of  a  1st,  2d,  . ...kth 
favorable  case,  also  of  a  1st,  2d,  ....rth  unfavorable  case,  and  the  product 
expressing  the  probability  of  these  occurring  in  a  certain  order. 

EXAMPLE. 

If  from  20  tickets,  8  of  which  are  prizes  and  12  blanks,  6  are  to  be  drawn, 
then,  in  favor  of  the  requisition  that  exactly  two  prizes  shall  be  first  drawn,  or 
shall  occupy  any  given  place  in  the  order, 

77 


W 


~  \20/  U9/  X  \18/  U7/  \16/  \15/  — J 


"3230 

V.  Should  there  be  required  in  the  supposition  of  the  last  case  no  particu- 
lar order  for  the  single  cases  which  occur,  the  expression  becomes 

k+I\m/'"\i7i—k+l/'\m—k/'"\7n—k—r+ir  '  '  '  v    '; 
Thus  it  will  be  found  that,  if  from  30  appointed  numbers  out  of  90,  5  of  the 
whole  90  are  to  be  drawn,  so  that  just  3  of  the  30  shall  be  among  those  drawn, 
it  being  immaterial  at  which  three  of  the  five  drawings,  the  expression  for  the 
probability  in  this  case  is 

/5.4.3\    /30\/29\/28\    /60\  /59\       20650 
~  \l.2.3/  "  \907  \89/  \88/  '  \87/  \867  ""126291' 

VI.  Should  the  number  of  possible  cases  continue  to  remain   the  same, 
while  the  other  circumstances  are  as  in  (V.),  the  formula  would  be 

C   (g-Y.W. (VI.) 

K+rW/       \Mi/  v  ' 


251  ALGEBRA 

EXAMPLE. 

The  probability  of  throwing  the  same  lace  three  times  in  7  casts  of  a  die, 
or  one  cast  of  7  dice,  would  be  expressed  l>y 


7.6.5/l\:i  (5\*_  21875 
1.2.3\0V  '\6/  :  >J36' 


VII.  Let  the  probability  be  required  that  of  two  different  occurrences  the 
first,  or,  if  this  does  not,  the  second,  shall  happen  ;  if  the  single  probability  of 
the  lir-~l  happening  be  expressed  by  w,  the  probability  of  its  failing  wilf  be  ex- 
pressed by  1—  iv ;  this  must  be  combined  with  the  probability  of  the  second 
happening,  according  to  (III.),  giving 

(1  — «'i)w2 
for  the  probability  of  the  second  happening,  if  the  first  fails  :  then  the  com- 
pound probability  required  is  expressed  (II.)  by 

W=:Wl-\-Wa(l—W1)=Wl+Wa—Wl  .W2. 

EXAMPLE. 

Required  the  probability  of  throwing  with  two  dice,  at  the  first  cast  8,  and, 
if  this  does  not  happen,  9  at  the  second  cast. 

w=—*  4--  (l  -—)—— J_i-  -—™ 

VIII.  Above  we  have  considered  the  absolute  probability  of  the  happening 
of  an  event ;  the  relative  probability  of  the  happening  of  two  events  is  ex- 
pressed by  the  formula 

— i — '  or  — r — • 

Wl-\-W2  M,-|-«!, 

EXAMPLE. 

The  relative  probability  of  throwing  with  two  dice  rather  7  than  10,  is  ex- 

.  .  «>i  6         2 

pressed  by 


Wi+w*  6+3~3' 
IX.  When  money  depends  on  the  happening  of  an  c. .  it,  the  product  of 
the  sum  risked,  multiplied  by  the  expression  for  the  probability  of  the  event 
on  which  it  depends,  is  called  the  mathematical  expectation.  If  there  be 
among  m^nii  cases,  nii  favorable  for  one  pan  v.  and  >n:  for  tho  other,  the 
sum  risked  by  the  first  a„  and  by  the  second  ,r,  then  for  tho  mathematical 
expectation  of  each  we  have 

Therefore,  when  r[=r:,  it  is  necessary  that  -/,  :  a:=u\  :  w:.  This  principle 
is  important  in  the  subject  of  annuities  and  life  insurance.  For  its  nppl  eation, 
and  that  of  all  tho  foregoing  theory  to  which,  see  De  Morgan  on  Probabilities. 

r\  vm  PLES. 

(1)  How  many  binary  combinations  of  oxygen,  hydrogen,  nitrogen,  carbon, 

sulphur,  and  phosphorus?      I  low  many  ternary  combinations  of  the  same.' 

(2)  llow  many  combinations  of  5  colors  among  those  of  the  prism,  via.,  red, 

orange,  yellow,  green,  blue,  indigo,  and  violet  .' 

19  and  9  can  each  be  thrown  with  two  dice  bol  La  one  way,  it  nn<l  3  each  in  two 
ways,  to  and  i  in  throe  ways,  r>  and  'J  in  bar  ways,  G  and  h  in  live  ways,  7  in  six  ways. 


MKTIIOD  OF  UNDETERMINED  COEFFICIENTS.  255 

(3)  What  is  the  probability  of  throwing  with  three  dice  two  equal  num- 
bers 1  wilii  five  dice,  three  equal  ? 

(4)  What  of  throwing  with  two  dice  the  faces  2,  4,  and  6  ? 

(5)  What  the  probability  lhat  a  dollar  tossed  twice  will  fall  head  up  onco  ? 

(6)  Of  which  is  the  probability  greater,  the  drawing  at  three  trials  from 
52  cards  three  cards  of  different  colors,  of  which  there  are  four,  or  three  face 
cards,  of  which  there  are  12  ? 

(7)  What  of  drawing  out  of  a  vase  containing  5  white,  6  red,  and  7  black 
balls,  in  two  drawings,  2  red,  or  else  a  white  and  a  black  ball  ? 

(8)  What  of  drawing  out  of  the  same  vase,  in  three  drawings,  3  of  differ- 
ent colors,  or  else  2  black  and  1  white  ? 

(9)  What  of  throwing  with  four  dice  15,  or  with  three  dice  12  ? 

METHOD  OF  UNDETERMINED  COEFFICIENTS. 
209.  The  method  of  undetermined  coefficients  is  a  method  for  the  expan- 
sion or  development  of  algebraic  functions  into  infinite  series,  arranged  accord- 
ing to  the  ascending  powers  of  one  of  the  quantities  considered  as  a  variable.* 
The  principle  employed  in  this  method  may  be  stated  in  the  following 

THEOREM. 

If  A.ra-fB.r/i-f-C.ry+,  &c,  =A'.r"'  +  B'.ni'+C '.t*'+,  &c.  (1),  for  all  values 
of  x,  then  must  the  exponents  of  a:  in  the  two  members  be  the  same,  and  the  co- 
efficients of  the  same  powers  of  .r  the  same.    For,  dividing  (1)  by  x",  we  have 
A  +  B-rP— »  +  C.r>'-«  +  ,  &c.,  r=A'.ra-a4-B'.r/3'-^+C'.r>'-«-|-,  &c.  (2) 

Since  x  may  have  any  value,  make  it  zero;  the  first  member  thus  reduces 
to  A,  while  tl«  second  becomes  zero,  unless  we  suppose  a  equai  to  some  one 

of  the  exponents  a',  /?',  y', Suppose  it  to  be  a'.     Then  we  have  «  =  a\ 

and  .-.  A  =  A'.     Suppressing  the  equal  terms  A  and  A'x*'—<*  from  the  two 
members  of  (2),  and  dividing  it  by  xP— ■<,  it  becomes 

B-|-C.r>'-^+,  &c,  =r.'.r,.;-,3+C.r>'-.3+,  &C 

Making,  again,  ,r=0,  the  first  member  reduces  to  B,  and  the  second  to  zero, 
which  is  absurd,  unless  we  make  ,6'  equal  to  some  one  of  the  exponents  of  x, 
say  /?',  in  tne  second  member,  and  then  13  =B'.  Proceeding  in  this  way,  the 
exponents  of  .r,  and  the  coefficients  of  the  same  powers  of  x  in  the  one  mem- 
ber, may  be  proved  equal  to  those  in  the  other. 

The  above  theorem  may  be  expressed  in  a  modified  form ;  thus,  if  all  the 
terms  of  (1)  be  transposed  to  the  first  member,  it  becomes,  collecting  the  equal 
powers  of  x,  a  and  a',  /?  and  j3',  &c, 

(A— A').r«  +  (B  —  B').r£+(C  —  C').rr+,  &c,  =0; 
from  which,  since  A  =  A',  B=B',  &c,  we  perceive  that  when  a  function  of 
x  is  equal  to  zero  for  all  values  of  .r,  the  coefficients  of  the  different  powers  of 
x  are  equal  to  zero  separately. 

EXAMPLES. 

(1)  Expand  the  fraction — ; — :  into  an  infinite  series. 

v   '         *  1 — 2.f+.ra 

Assume  1_„         ;=A4-B.r-f-C.r-  +  P.i.-"-f-E.^+  . . . .  , 

*  A  variable  quantity  is  one  which  is  either  entirely  indeterminate,  so  that  it  may  have 
any  value  at  pleasure,  or  one  which  varies  in  conformity  with  certain  conditions  imposed. 


ALGEBRA. 

in  which  some  of  the  coeffic  •  B,  C,  &<■-,  a  sei  >,  and  thus  certain 

powers  of  x  be  wanting;  then,  mull  y  '  —  ;,Jr'  •  w':  hwi 

i=A+  Bx-f.  Cj  +   I-*    -h  i.  H 

—  .-A/  —  >B         '     —2D    — .... 
+  A/ +  J^.;+   Cz*4-.... 
Hence,  by  the  preceding  theorem,  we  I 

A  =  l  .-.  A=     ...    =1 

B— 2A=0      B=;\         =2 

C— 2B+A=sO      C=2B— A=3 

D— 2C-fBs=0      Ds=2C—B=4 

E— 2D+C=0      E=2D— C=5 

&c.  &c. 


Therefore       x _ .-, x  ■   ,., =  1  +  2x + 3x'-  +  4 /■  -f  ', /."  +  W  -f 

Tlie  equality  of  a  function  to  a  series  is  hypothetical ;  and  after  A,  B,  C 
have  been  found,  the  result  must  be  carefully  examined.     If  we  put  the  func- 
tion ; =A-r-Bz-j-,  &c,  it  gives  the  absurdity  — 1=0.     We  mu 

Az-l  +  Bx°+Cr+D.r!-|-,  &c.    The  method  of  indeterminate  coeffi- 


3x— Xs 

cients  is  to  be  avoided  where  other  methods  will  apply. 

(2)  Extract  the  square  root  of  1-j-x.  * 

Assume  V l+x=A  +  Bx    -4-C./.:    +Dx')-f- ...,  and  square  both  sides; 
.•.l+x=A2+ABx+AC<  +AI>    +AEx»+.. 
+  ABx+B;x-  4-BCx»+BDx«+.. 

+  ACx-  +  BCx-+C:x<   +.. 

4-ADx';+BDr*4-.. 

+  AEx«+.. 

Hence,  equating  the  coefficients  of  the  like  powers  of  x,  we  have 

A  =1  .-.A=2       1 

1  1  1 

2AB=1      B=      -^=     -=     § 

2AC  +  B—0       C  =  -^-=-  — =-- 

\'.r  i  i 

2AD4-2BC=0      D=— r-=     —  =   — 

A  J  -         16 

2BD+I  1  <;  1       l  >  5 

2AE  +  2BD  +  C-0      E  =  — £-    =  -,  j  -  +  -  \  =--, 

Ate. 


Therefore  \Zl  +  .<  —  i(l  +  -A-r— Jj--+ ,'      -  +....)■ 

3x— ."> 
(3)  Decompose  -= — — — t—tt  into  two  I  simple  binomial  d«*- 

x  '  '         xa — lox-|-10 

nominators. 

By  quadratics  we  find  X* —  13x+40  =  (r — 5)(x—  ace  we  may  assume 

3x— 5  A       _B A(x— 8)+B     -        (A+B)x— 8A— 5B 

x*_i;jx+40=x— 5+x— 8—      (x— 5)(x-  x*— 13x-f40        ' 

.-.  ::,—  .-)  =  (A  +  B)r  — 
and  by  thn  principle  of  undetermined  coefficients  we  have 

A  +  B  =  3,  and  8A+5B  = 


: 


- 


' 


I- 


— 


•"I  -        —  •     " 


1 


. 


5  ;"_ 


. 


258  ALGJSBHA 

a  —  bx 
(8)  Exrand  — - — -  to  four  terms. 
a-\-cx 


Ans.  l-(b+c)l+c{b+c£-c"ib+r  £+. 


x-}-2 
(9)  Resolve  — ; ■  into  partial  fractions. 


a  J_     ,     _3_       2 

Ans'2(x+l)  +  2|x-l)~x 


(10)  Resolve  -r- .,,.,   , — .  into  partial  fractious. 

'  .?.'J(1 — x)-(l-f-r) 

112  1 


X*     X*     X     '2(1— x)*  '  4(1— .r,      4^1+x^ 


(11)  Expand  x>+2ax+ai  to  four  terms 


2a     3a*     4a3 
Ans.  1—  -| — r— 3-+ 

XX"  Xs     ' 


(12)  Resolve  — — -  into  partial  fractions. 

Ads. 


1  1  1 

4(x—l)~4(x+l)— 2(^+1)' 


LOGARITHMS. 

210.  Logarithms  are  artificial  numbers  adapted  to  natural  numbers,  in 
jrder  to  facilitate  numerical  calculations ;  and  we  shall  now  proceed  to  explaiu 
the  theory  of  these  numbers,  and  illustrate  the  principles  upon  which  their 
properties  depend. 

Definition. — In  a  system  of  logarithms,  all  numbers  are  consl  i  the 

powers  of  some  one  number,  arbitrarily  assumed,  which  is  called  the  base  of 
Oie  system,  and  the  exponent  of  that  power  of  the  base  which  is  equal  to  any 
given  number  is  called  the  Logarithm  of  that  number. 
•      Thus,  if  a  be  the  base  of  a  system  of  logarithms,  N  any  number,  and  x  such 
that 

Nacax, 
then  X  is  called  the  logarithm  of  N,  in  \\  e  Bystem  whose  base  is  a. 

The  base  of  the  common  system  of  logarithms  (called,  from  their  inventor, 
"Briggs's  Logarithms")  is  the  number  10.     Hence,  since 

(10)°=  1,  0  is  the  logarithm  of  1  in  this  system, 
(!())'=  10,  1  is  the  logarithm  of  10  in  this  system, 
(10)-=  100,  2  is  the  logarithm  of  100  in  this  system, 
(10):,=  1000,    .".  is  the  logarithm  of  1000   in  this  system, 

(10)4±10000,  I  is  the  logarithm  of  10000  in  tins  system, 
flee.  =     dec.     iVc 

211.  In  crder  to  havo  the  numbers  corresponding  to  the  logarithms  L,  j  or 
0.5,  I  or  0.»5,  &c,  it  is  necessary  to  extract  the  Bquare,  4th,  and  so  on,  root 
of  10,  or  tt  ttttct  the  square  root  successively,  as  exhibited  in  die  following 
tuble  : 


LOGARITHMS 


259 


Nnm.-er  of  times  that  llie 
square  root  ia  extracted 
successively. 

Exponent* 
Numbers.                                                                         or 

Logarithms. 

0 

1 

2 
3 
4 
5 
6 

10,000  0000 
3,162  2777 
1,778  2794 
1,333  5214 
1,154  7819 
1,074  6078 
1,036  6329 

1,000  0000 
0,500  0000 
0,250  0000 
0,125  0000     • 
0,062  5000 
0,031   2500 
H.Ol.-,   U250 

7 

8 

9 

10 

11 

12 

1,018  1517 
1,009  0350 
1,004  5073 
1,002  2511 
1,001   1249 
1,000  5623 

0,007  8125 
0,003  9062 
0,001  9531 
0,000  9765 
0,000  4882 
0,000  2441 

13 

14 
15 
16 

17 

18 

1,000  2811 
1,000  1405 
1,000  0702 
1,000  0351 
1.000  0175 
1,000  0087 

0,000   1220 
0,000  0610 
0,000  0305 
0,000  0152 
0,000  0076 
0,000  0038 

19 
20 
21 
22 
23 
24 

1,000  0043 
1,000  0021 
1,000  0010 
1,000  0005 
■1,000  0002 
1,000  0001 

0,000  0019 
0,000  0009 
0,000  0004 
0,000  0002 
0,000  0001 
0.000   oooo 

By  means  of  the  above  table,  to  calculate  the  logarithm  of  any  number  (A) 
between  1  and  10  accurately  to  5  places  of  decimals,  take  out  from  the  second 
column  the  nearest  number  to  A,  but  less,  and  divide  A  by  this.  Take  out, 
again,  the  next  logs  number  than  the  quotient  B,  as  a  divisor  for  B,  and  so  on 
until  the  last  quotient  contains  only  millionths  ;  the  logarithm  sought  is  the 
sum  of  all  the  exponents  or  logarithms  in  the  third  column  corresponding  to 
the  divisors  used  from  the  second.  For,  calling  these  exponents  a,  (3,  y,  6 
we  have 

A=B;i   c    c_D   D 

10a  10"  10y  10d 

.-.  A=10aB  =  10a  X  10^C  =  10a  X  10^  X  10yD=10a.  10^.  10y.  10*5. . . 

...A  =  lOa+0+}+5-. 

Any  exponent  beyond  6  being  added  to  the  others  would  not  affect  the 
millionth  place,  or  fifth  decimal.  Q.  E.  D. 

Now,  inasmuch  as  all  numbers  lying  between  the  1st,  2d,  3d,  &c,  powers 
of  10  must  have  broken  numbers  for  logarithms,  these  numbers  will  be  of  the 

.  k  a         k_ 

form  10  mr=10  .10m  ;  hence  the  calculation  of  their  logarithms  will  in  every 
case  depend  on  the  calculation  of  a  fractional  logarithm  such  as  has  been  just 
exhibited. 

A  table  of  logarithms  is  a  table  containing  all  numbers  from  1  up  to  10000 
or  100000,  or  some  high  number,  with  their  corresponding  logarithms. 

These  tables  are  made  with  certain  abbreviations  and  conveniences,  which 
we  shall  presently  explain. 

From  the  scheme  of  numbers  in  (210)  it  appears,  that  in  the  common  sys- 
tem the  logarithm  of  every  number  between  1  and  10  is  some  number  between 


260  ALGEBRA. 

0  and  1,  i.  e.,  is  a  fraction.  The  logarithm  of  every  .number  between  10  and 
100  is  some  number  between  1  and  2,  i.  c.,  is  1  plus  a  fraction.  The  logarithm 
of  every  number  between  100  and  1000  is  some  number  betwoen  2  and  3,  i-  < •  , 
is  2  plus  a  fraction,  and  so  on.  The  whole  Dumber,  or  integral  part  of  the 
logarithm,  is  called  the  index,  or,  more  commonly,  the  characteristic. 

212.  In  the  common  tables  of  logarithms  the  fractional  part  alone  of  tht> 
logarithm  is  registered,  and  from  what  has  been  said  above,  the  rule  usually 
given  for  finding  the  characteristic,  or  index,  will  be  readily  understood,  vi/.  : 
The  index  of  the  logarithm  of  any  number  greater  Otan  unity  is  equal  to  ont 
less  than  tfie  number  of  integral  figures  in  Oie  given  number;  for  if  the  num 
ber  be  between  10  and  100,  it  will  contain  two  integral  figures  ;  if  between  100 
and  1000,  it  will  contain  three,  and  so  on.  Thus,  in  searching  for  the  logarithm 
of  such  a  number  as  2970,  we  find  in  the  tables  opposite  to  2970  the  number 
4727564;  but  since  2970  is  a  number  between  1000  and  10000,  its  logarithm 
must  be  some  number  between  3  and  4,  i.  e.,  must  be  3  plus  a  fraction  ;  the 
fractional  part  is  the  number  47275G4,  which  we  have  found  in  the  tables ; 
prefixing  to  this  the  index  3,  and  interposing  a  decimal  point,  we  have  3.47275G4, 
the  logarithm  of  2970. 

We  must  not,  however,  suppose  that  the  number  3.4727564  is  the  exact 
logarithm  of  2970,  or  that 

2970  =  (10)3-,"75<* 

accurately.  The  above  is  only  an  approximate  value  of  the  logarithm  of  2970 , 
we  can  obtain  the  exact  logarithms  of  veiy  few  numbers;  but,  taking  a  sufficient 
number  of  decimals,  we  can  approach  as  nearly  as  we  please  to  the  true 
logarithms. 

213.  It  has  been  shown  that  in  Briggs's  system  the  logarithm  of  1  is  0  ;  con- 
sequently, if  we  wish  to  extend  the  application  of  logarithms  to  fractions,  we 
must  establish  a  convention  by  which  the  logarithms  of  numbers  less  than  1 
may  be  represented  by  numbers  less  than  zero,  i.  <?.,  by  neg  /mbers. 

Extending,  therefore,  the  above  principles  to  negative  exponents,  since 

or  (10)_1=0.1,         — 1  is  the  logarithm  of  .1         in  this  system, 


10 

1 
100 

1 

1000 

1 


or  (10)~5=0.01,  — 2  is  the  logarithm  of  .01  in  this  system, 
or  (10)_3=0.001,  —3  is  the  logarithm  of  .001  in  this  system, 
or  (10)_4=0.0001,  —4  is  the  logarithm  of  .0001  in  this  system, 


10000 

&c  &c. 

It  appears,  then,  from  this  convention,  that  the  logarithm  of  every  number 
between  1  and  .1  is  some  number  between  0  and  — 1  ;  the  logarithm  of  every 
number  between  .1  and  .01  is  some  number  between  — l  and  — 2;  the 
logarithm  of  every  number  between  .01  and  .001  is  some  number  between 
— '.'  and  — .'!,  and  so  on. 

From  this  will  bo  understood  tho  rulo  given  in  books  of  tallies  for  finding 
the  characteristic,  or  index,  of  tho  logarithm  of  n  decimal  fraction,  viz. :   The  in- 
form decimal  fraction  is  a  negative  number,  equal  to  unity,  added  to  tltf 
number  of  zeros  immediately  following  the  decimal  point.    Thus,  in  searching 
for  a  logarithm  of  tin-  number  such  as  .00462,  we  find  in  the  tables  opposite  to 


LOGARITHM*.  26  J 

462  the  number  6646420;  but  since  .00462  is  a  number  between  .001  and  .01 
its  logarithm  must  be  some  number  between  — 3  and  — 2,  i.  c,  must  be  — 3 
^lus  a  fraction;  the  fractional  part  is  the  number  6646420,  which  we  have 
found  in  the  tables;  therefore  —  3-f-'6G40420  is  the  logarithm  of  .00462.  It 
is  customary  to  write  the  sign  —  over  the  characteristic  to  show  that  it  affecta 
that  alone,  and  not  the  decimal  part  of  the  logarithm,  which  is  positive  ;  thus, 
3.6646420. 

GENERAL  PROPERTIES  OF  LOGARITHMS. 

214.  Let  N  and  N'  be  any  two  numbers,  x  and  x'  their  respective  logarithms* 
a  the  base  of  the  system.     Then,  by  definition, 

N  =a* (1) 

N'=a"' (2) 

I.  Multiply  equations  (1)  and  (2)  together, 

=  ax+x' 
.-.  by  definition,  x-\-x'  is  the  logarithm  of  NN' ;  that  is  to  say, 
The  logarithm  of  the  product  of  two  or  more  factors  is  equal  to  the  sum  of  the 
logarithms  of  those  factors. 

II.  Divide  equation  (1)  by  (2). 

N      a* 
N'=aT' 


-x> 


=ax 
.-.  by  definition,  x — x'  is  the  logarithm  of  ^r, ;  that  is  to  say, 

The  logarithm  of  a  fraction,  it  of  the  quotient  of  two  numbers,  is  equal  to  the 
logarithm  of  the  numerator  minus  the  logarithm  of  the  denominator. 

III.  Raise  both  members  of  equation  (1)  to  the  nth  power. 

Nn=a"". 
.-.  by  definition,  nx  is  the  logarithm  of  Nn ;  that  is  to  say, 
The  logarithm  of  any  power  of  a  given  number  is  equal  to  the  logarithm 
of  the  number  multiplied  by  the  exponent  of  the  power. 

IV.  Extract  the  ntil  root  of  both  members  of  equation  (1). 

1  X 

N7l=a7). 

x  I 

.-.  by  definition,  -  is  the  logarithm  of  N" ;  that  is  to  say, 

The  logarithm  of  any  root  of  a  given  number  is  equal  to  the  logarithm  of  the 
number  divided  by  the  index  of  the  root. 

Combining  the  last  two  cases,  we  shall  find 

m  n\x 

N~°=ef", 

mx  ™ 

whence  — -  is  the  logarithm  of  N". 

n  ° 

It  is  of  the  highest  importance  to  the  student  to  make  himself  familiar  with 
the  application  of  the  above  principles  to  algebraic  calculations.  The  following 
examples  will  afford  a  useful  exercise  : 

(1)  Log.  {a,  b,  :,d )=  log.  a-\-  log.  b+  log.  c+  log.  d.... 

(2)  Log.  (^)  =  log.  a+  log.  5+  log.  c—  log.  d—  log.  e. 


26-2  ALGEBRA. 

(3)  Log.  (a'nb"cv )=m  log.  a-f  n  log.  b+p  lug.  c... 

(4)  Log.  \-^r)  =m  log.  a  +  n  log.  £>— J>.  log.  c. 

(5)  Log.  (a2— :r2)=log  (a+ar)x(a—  x)=  log.  («/  +  .r)  +  log-  («*-*)• 
1  .  1 


(6)  Log.  V"-'—  i'-  =  o  loS-  ("+0  +  .J  ^g-  ("—■'•)• 

1  3  15 

(7)  Log.  (a8V<^)=  log-  fl3+7  log-  "3=3  loS-  a  +  4  lo§-  a  =  7  loS-  a- 

m  m 

(8)  Log.  V(a»-ar>)»=-  log.  (a-*)+-  log.  (d»+ax+i») 

=— {log.  (a—a:)4- log.  (a+a:+z)+ log.  (a+z—z) J 

where  z-=ax. 

(9)  Log.  V«Q+^-=oilog-  (a+x+:)+  lo§-  (a+x  — =)}»  where  ;2  =  2a.-> 


(10)  Log-   ^!{_x)!=2ll0S-  («-*)— 8  '°g-  («+*)!• 

TABLES   OF  LOGARITHMS. 

mhe  principal  French  tables  are  those  of  31.  Callet,  an  American  edition  ot 
which  has  been  made  by  the  late  Mr.  Hasler.  The  first  of  these  table9 
marked  Chiliade  I., occupying  only  five  pages,  contains  the  series  of  numbers 
from  1  up  to  1200,  with  their  logarithms  Expressed  to  eight  places  of  decimals, 
the  numbers  Being  in  the  column  marked  N,  and  their  logarithms  in  the  column 
marked  Log.*  The  second  table,  which  is  of  far  greater  bulk,  exhibits  the 
logarithms  of  all  entire  numbers  from  1020  up  to  10800.  The  numbers  are  in 
the  column  entitled  N,  and  their  logarithms  in  the  following  column,  marked  0. 
The  characteristics  of  the  logarithms  are  not  written  in  the  tables,  since  they 
may  be  known  without,  being  always  one  less  than  the  Dumber  of  dibits  of 
which  the  number  to  which  the  logarithm  belongs  is  composed.  The  logarithms 
of  numbers  containing  one  figure  more  than  those  in  the  column  N,  are  found 
by  means  of  the  columns  marked  at  top  1,  2,  3,  ...  0.  Thus,  to  find  the 
logarithm  of  27796,  seek  in  the  column  N  the  number  2779;  run  along  the 
horizontal  line  which  contains  this  number  to  the  column  marked  i> ;  yon  find 
there  the  last  four  figures  of  the  logarithm  sought :  the  first  three  figures  of  it 
are  found  in  the  column  marked  0,  t>>  the  left  of  the  period,  on  the  Bame 
horizontal  line,  or  a  little  above.  You  obtain  thus,  after  prefixing  the  proper 
characteristic, 

log.  27796=1.11.;" 

It  will  be  seen,  by  inspecting  the  tables,  that  the  diiVerences  of  the  consecu- 
tive logarithms  is  constantly  the  same  for  b  considerable  number  of  them,  and 

as  the  diH'eivntes  of  the  consecutive  numbers  is  also  constant,  it  follows  that 

*  This  table  also  miliums  an  arrangi  intent  tor  reducing  uiiiiutfs  and  aeoooda  to  aacoada 
without  the  trouble  of  multiplying  by  60.    Tims,  on  the  fourth  page,  we  find  i-'  in  tin 
of  tlio  columns  marked  I"--,  and  against  30,  in  the  Brat  column  marked  ".  we  Bad  740, 
which  is  the  Dumber  of  seconds,  in  i~'  80".    By  this  arrangement  we  find  readily  tho 
I         tlim  of  the  seconds  in  aj  n  Dumber  of  minutei  and  seconds,  which  i* 

nt  in  astronomical  calculations.    It  is  evident  that  theae  numbers  might  be  J 

as  degrees  and  minutes,  or  hours  and  minutea,  as  well  as  annates  and  seconds 


LOGARITHMS.  263 

the  differences  of  the  logarithms  are  proportional  to  the  differences  of  the 
numbers.     Suppose,  then,  that  the  logarithm  of  14518469  were  required. 

From  the  tables  we  find,  as  before,  neglecting  for  the  present  the  charac- 
teristic (see  a  page  of  the  tables  of  Callet  at  the  end  of  this  volume), 

log.  14518  =  1619008. 

This  is  also  the  logarithm  of  14518000,  which  differs  from  the  logarithm  of 
the  next  number  14519,  or  14519000,  viz.,  1619367  by  299,  while  the  num- 
bers themselves  differ  by  1000.  But  the  number  14518000  differs  from  the 
given  number  14518469  by  469,  the  last  three  figures  not  yet  used  ;  hence 
the  proportion 

Dif.  Nos.  Dif.Logs.   DiCNos.        Dif.ofLogs. 

1000  :  299  ::  469  :  x=141, 

which  result,  added  to  1619068,  gives  7.1619209  for  the  logarithm  required,  7 

being  the  proper  characteristic  for  the  logarithm  of  a  number  consisting  of 

eight  figures. 

299 
The  proportion  is  solved  by  multiplying  the  difference  469  by  ttjtjt:,  or  by 

2        9  9 

— 4- -J- .     Now,  by  inspecting  the  last  column  of  the  page,  this  differ 

10^100^1000  »    J       f         s  it.' 

ence,  299,  will  be  found  ready  calculated,  and  its  product  as  nearly  as  it  can  be 

12       3 
expressed  injiwo  or  three  figures  by   — ,   — ,   — ,  &c,  or  .1,  .2,  .3,  &c,  the 

multiplier  being  in  the  left  hand  and  the  product  in  the  right  hand  of  the  two 
small  columns  of  figures  under  the  difference,  299.  These  multipliers  may  be 
regarded  as  hundredths  or  thousandths,  only  giving  the  products  their  proper 
place.     With  this  explanation,  the  following  calculation  will  be  understood  : 

Log.  14518  1619068 

0.4       120 

0.06     18* 

0.009 3 

Log.  14518469 7.1619209 

215.  To  find  the  number  corresponding  to  a  given  logarithm,  say  1619209, 
look  in  the  column  marked  0  for  the  nearest  less  logarithm,  and  take  the  cor- 
responding number,  which  is  1451.  Run  the  eye  along  the  horizontal  line  till 
the  number  most  nearly  approaching  9209,  forming  the  last  four  figures  of  the 
given  logarithm,  is  found.  This  is  9068,  which  is  found  in  column  8.  Sub- 
tract  this  from  9209,  and  the  difference  is  141.  Find  in  the  right  hand  of  the 
two  columns  of  small  figures  marked  dif.  et  p.,  or  simply  dif.,  at  the  top  of  the 
page,  the  nearest  less  number  than  141 ;  this  is  120,  which  answers  to  4  in 
the  left  hand.  The  difference  between  120  and  141  is  21.  Multiply  21  by 
10,  and  seek,  as  before,  in  the  small  column,  the  number  nearest  210 ;  this  is 
209,  which  answers  to  7.  The  calculation  is  below. 
Log.  .r=1619209 

For  1619068 14518 

First  remainder,  141 04 

Second  remainder,        21 007 

a-=  1451347.    fc 
The  numbers  4  and  7  thus  found  may  be  simply  annexed  to  14518. 

*  The  number  in  the  table  is  179  ;  but,  us  the  9  is  rejected,  the  7  is  increased  by  1,  sinew 
179  is  nearer  180  than  170. 


264  ALGEBRA. 

If  the  characteristic  of  the  logarithm  had  been 

6,  the  number  would  have  been  1451847  ; 
5,  the  number  would  have  been  145184.7  ; 
4,  the  number  would  have  been    14518.47  ; 
1,  the  number  would  have  been  1 1.51847; 

0,  the  number  would  have  been  1.451847; 

1,  the  number  would  have  been  .1451847  ; 

2,  the  number  would  have  been  .01451617. 

This  table  contains  in  tho  first  three  columns  an  arrangement  for  reducing 
any  number  of  degrees,  minutes,  and  seconds,  or  hours,  minutes,  and  seconds, 
to  seconds,  which  is  particularly  useful  in  astronomical  calculations,  where  the 
logarithm  of  the  number  of  seconds  in  a  given  number  of  degrees,  minutes,  and 
seconds  is  frequently  required. 

EXAMPLE   I. 

Reduce  0°  or  0b  24'  57"  to  seconds.  In  the  table  (see  last  page),  at  the 
head  of  the  first  column,  find  0°,  and  immediately  under  it  24' ;  descending 
this  column  to  55",  near  the  bottom,  and  opposite  57",  which  is  understood  to 
be  two  numbers  below,  is  found  1407,  the  number  of  seconds  required. 

If  the  degrees  or  hours  exceed  3,  the  proceeding  is  different. 

EXAMPLE   II. 

To  reduce  4°  or  41'  2'  39"  to  seconds.  Find  4°  0'  at  the  head  of  the 
second  column,  and  below,  in  this  same  column,  2'  30",  to  which  corresponds, 
in  the  third  column,  1455.     Thus,  4°  2'  30"  =  14550"  .-.4°  2'  39"  =  14559" 

EXAMPLES   OF   THE  APPLICATION   OF  LOGARITHMS. 

(1)  To  find  the  value  to  within  0.01  of  the  expression 

_  7340X3549 
r=G81.8x  593.1' 
By  the  properties  of  logarithms, 

log.  x=  log.  7340-f  log.  3549—  log.  681.8—  log.  593.1. 
The  following  is  the  calculation : 


log.  7340=3.8656961 

log.  3549=3.5501060 

sum  =7.4158021 


log.  681.8=2.8336570 

log.  593.1  =2.7 731279 

sum  =  .j.b'Uu'7649 


First  sum,  =7.4158021 
Second  sum,  =5.60678 49 
Dift*.  or  log.  .r= 1.8090172 

216.  The  arithmetical  complement  of  a  logarithm  is  what  remains  after  the 
logarithm  is  subtracted  from  10.  Thus,  the  arithmetical  complement  of  the 
logarithm  2.7190826  is  10— 2.7190826=7.280917  1.  which  ifl  obtained  by  be- 
ginning on  the  right  and  subtracting  each  figure  (carrying  1  to  all  except  the 
Bret)  from  10,  or  beginning  on  the  left  and  subtracting  each  figure  of  tho 
logarithm  from  9,  except  the  last,  which  is  subtracted  from  LO. 

217.  Thojaperation  of  subtraction  of  logarithms  can  be  replaced  by  addition, 
if  wo  use  the  arithmetic  complement  ;  for  n'.  to  a  given  logarithm,  log.  <;,  wo 
add  the  arithmetical  complement  of  another  logarithm,  such  as  10—  lug.  6 
we  have 


LOGARITHMS.  265 

log.  a  -f- 10  —  log.  b, 
from  which,  rejecting  10,  the  result  is 

log.  a— log., 6, 
the  same  as  would  be  obtained  by  simply  subtracting  the  second  logarithm 
from  the  first. 

We  have  then  the  following  rule  for  operating  with  arithmetical  comple- 
ments :  Add  the  arithmetical  complements  of  the  logarithms  of  the  divisors  and 
the  logarithms  of  the  multipliers  of  a  formula  together,  rejecting  10  from  the 
sum  for  every  arithmetical  complement  employed. 

The  above  example  would  be  wrought  by  this  rule  as  follows : 

log.  7340  =  3.8656961 

log.  3549=3.5501060 

ar.  comp.  log.  681.8=7.1663430 

ar.  comp.  log.  593.1=7.2268721 

sum  rejecting  20=T78090172=log.  x, .-.  a:=64.42. 

We  thus  obtain  the  same  result  as  by  the  other  method.  The  number  cor- 
responding need  be  taken  from  the  tables  only  to  four  figures,  because,  the 
characteristic  being  1,  the  entire  part  of  the  number  will  contain  but  two 
places,  which  will  leave  two  places  for  the  decimal  part,  as  required,  since  the 
value  of  a:  was  to  be  obtained  to  within  0.01. 

(2)  To  find  the  value  within  0.00001  of  the  quotient. 

(•v/146298)4 
x=- 


(V988789)5 
By  the  rules, 

log.  ar= flog.  146298— flog.  988789, 
and  the  calculation  will  be  as  follows  : 


flog. 

146298. 

1  log- 

988789. 

log.  14629 

0.1652146 

log.  98878 

.  .  .    0.9950997 

for           0.8    . 

238 
5.1652384 

for            0.9    .  , 

...                  40 

log.  146298    . 

20.6609536 

.  .  .  29.9755185 

4.1321907 

.  .  .     4.9959197 

|log.  146298  =  4.1321907 

ar. 

comp.  flog.  988789=5.0040803 
sum  —10,  or  log.  x=1.1362710 

.-.  x=0 

.13686. 

/13 
(3)  Required  V/~  by  means  of  logarithms. 

13  log.  1.1139434 
27  log.  1.4313638 

11)1.6825796 

— 

V/— =.9357149  log.  1.9711436 


The  division  by  11  is  performed  by  adding  — 10  to  the  negati'e  part  of  the 
logarithm  and  +10  to  the  positive. 

The  logarithm  to  be  divided  is  viewed  as  if  written  thus  : 

—  114-10.66:25796. 


266  ALGEBRA. 

EXERCISES    IK    LOGARITHMS. 

(4)  Calculate  the  logarithm  of  8  from  the  table  on  page  259. 

(5)  Also  of  7,  70,  700,  7000,  70000. 

(6)  Also  of  356,  35G00,  3560000 

(7)  From  the  tables  find  the  logarithms  of  314,  3.721,  41.2. 

(8)  Also  of  7315,  8416,  91.75,  34760,  1708000. 

(9)  Find  the  numbers  the  logarithms  of  which  are  0.13130,  4.5651  ! 

(10)  Also  those  the  logarithms  of  which  are  3.6520528,  7.4891114. 

(11)  Those  the  logarithms  of  which  are  4.49010,  0.GG200,  5.72403. 

(12)  Find  by  proportional  parts  the  logarithms  of  314761,  440736,  37U2C-40U, 
2111768. 

(13)  Also  of  22.3345,  137.2014,  46.27835. 

(14)  Of  .75,  .341,  .7391,  .0347,  .000536,  .0000083. 

5   3    6    £  _7_ 

(15)  Of  -,  g,  —,  i3>  4Q- 

(16)  Find  the  logarithm  of  the  product  of  9.734  and  5.639. 

(17)  Also  of  35.98  X  7.433  X  6.543  X  29.78. 

(18)  Also  of  22.74  X  31.201  X  0.0067  X  0.9298. 

(19)  Divide  3758000  by  4986  by  means  of  logarithms. 

(20)  Also  16.87:0.07658  and  1.687:7658. 

(21)  Also  14.307:30415,  761.23:0.01871,  3.16:0.942. 

7      125     31       734        1 

(22)  Find  the  logarithm  of—,  — ,  — ,  — ,  — . 

(23)  Find  the  power  (5486)1  by  means  of  logarithms. 

(24)  Also  the  powers  (37.49)9,  (106.4)5,  (0.032)7,  (7.0034)8. 

/i\!b  /sy  /iy  /3\«  /i27\13 

(25)  Also  y    ,  {,)  ,  y  ,  (^  ,  [m)   . 

I  1\6      /  1\8      /  20\9      /  1     \3 

(26)  Also  (3+3)  ,  (4--)  ,  (7+-J  ,  (lOO-—;  . 

(27)  Find  the  cube  root  by  logarithms  of  1728000. 

(28)  Also  V34-782,  ^23990,  VC2S.73. 

11337       /9466       /120300     Kt 

(29)  Also  r^— ,  »]— ,  u/-^,  V0.1563,  ^0.0082. 

(30)  Also  y?;1-!^,  ^45390000,  V800.9. 

(31)  Also  V(1347)8,  7(70.44)",  V(8.664)». 

J/1722\5      //0.006\;i      j/72.93\7 

<32)  Also  vU  •  V  Wi  '  vfeos/  ■ 

(33)  Find  by  means  of  logarithms,  using  the  arithmetical  complement,  the 

27630X2678X5428 
valueof  36940  X  5302X7013" 

207.3  X  50.66  X  38.09  X  2713  X  0-098 

(34)  Also  of     344  x  o.763  x  0.4  X  6984  X7034.JJ    ' 

If  -85762X0.00853 

(35)  Also  of  y  7.58913X86.24  ' 

GAUSS    LOGARITHM-*. 

218.  The  common  logarithms,  or  logarithms  of  Briggs,  are  applicabls  only  to 
the  operations  of  multiplication,  division,  formation  ofpowera,  or  extraction  of 
roots,  and  do  not  apply  when  the  required  operation  is  that  of  addition  or  nib- 


LOGARITHMS.  liC7 

traction,  indicated  in  formulas  by  the  quantities  to  be  operated  upon  being  con- 
nected by  the  signs  -\-  and  — . 

A  system  of  logarithms  has,  however,  been  invented  by  Gauss,*  designed 
exclusively  for  sums  and  differences.  The  arrangement  of  these  tables,  which 
contain  three  columns,  marked  A,  B,  C,  is  founded  upon  tho  following  simple 
considerations. 

We  have  for  tho  form  of  a  sum  p-{-q,  and  of  a  difference  p — q,  tho  follow- 
ing identities : 

*+»(*£*) « 

p-(i=p-\~rq) (2) 

•••  log-  (p+q)=  iog..p+  log.  (^)  (3) 

and  log.  (p—q)=  \og.p—  log.  (j^r) 

The  logarithms  of  the  sum  p-\-q  and  the  difference^? — q  appear,  therefore, 
in  these  formulas,  equal  to  the  sum  or  difference  of  two  logarithms,  the  first 
of  which  is  to  bo  considered  as  directly  given,  but  the  second  of  which  must 
be  found  by  the  Gauss  tables.     They  contain, 


T.  In  the  column  A  logarithms  of  numbers  of  the  form  (-1,  increasing  from 
0.000  to  5.000. 

II.  In  column  B  logarithms  of  numbei's  of  the  form  ( J,  decreasing 

from  0.30103  to  0.00000. 

p-\-q 

III.  In  column  C  logarithms  of  numbers  of  the  form ,  increasing  from 

0.30103  to  5.00000. 

Now,  therefore,  inasmuch  as  log.  (~)=  log.  p —  log.  q,  by  the  tables  of 

common  logarithms,  the  first  thing  to  bo  done  is  to  take  the  difference  of  the 
common  logarithms  of  p  and  q,  enter  with  this  column  A  in  the  Gauss  loga- 
rithms, and  take  out  the  corresponding  number  from  column  B.  The  addition 
of  this  number  to  logarithm  p  will  give,  according  to  (3),  the  logarithm  sought 
ofp+q. 

In  order  to  find  the  logarithm  of  the  difference^ — q,  by  means  of  the  loga- 
rithms of  p  and  q,  two  cases  must  be  considered : 

/; 
1°.  Where  -<2  .•.  log. p —  log.  ^<0. 30103,  it  is  only  necessaiy  to  enter 

with  this  difference  column  B,  and  to  subtract  tho  adjoining  logarithm  of 
column  C  from  logarithm  p.     For,  corresponding  to  the  logarithms  of  numbers 

of  the  form  (— 1  in  B,  C  contains  the  logarithms  of  those  of  the  form  ( j. 

-  P 
2°.  If  —  >2  .-.  log.  p —  log.  5>0. 30103,  and,  therefore,   is   contained    in 

the  column  C  ;  subtract  the  corresponding  logarithm  in  column  B  from  loga- 
*  They  arc  found  in  the  latest  edition  of  the  tables  of  Vega,  and  those  edited  by  Kohler. 


268  ALGEBRA. 

P 
rithm  p  ;  because,  if  the  numbers  in  C  are  considered  =— ,  the  con  esponding 

P 

numbers  in  B  are 


p  —  q 

The  existence  of  the  foregoing  relations  between  B  and  C  is  easily  per- 
ceived if  we  substitute  in  II.  and  III.  the  value  p — 7  for  p,  and  afterward  q 
for  p — q. 

EXAMPLES. 

(1)  Let  log.  p=8.24502  and  log.  £  =  2.74194,  to  find  log.  (p  +  q).  We 
enter  column  A  with  the  log.  p —  log.  5  =  0.50308,  and  the  corresponding  log. 
in  column  B  =  0. 11861,  .-. 

log._p+B=3.24502-L.0.11801  =  3.36363=  log.  2310. 

(2)  From  log.  p=3. 32675  and  log.  7=2.09482,  to  determine  log.  (p—q)- 
Find  by  means  of  proportional  parts  for  the  value  of  log.  p —  log.  q  in  column 
B  the  corresponding  log.  in  C  =0.38325;  consequently. 

log.  p_C=3.32675— 0.38325=2.94350=  log.  878. 

(3)  From  log.  p=2. 64207  and  log.  7  =  1.87640  the  log.  of  (p—q)  is  found 
by  subtracting  from  the  nearest  value  of  log.  p —  log.  7=0.76567,  in  column 
C,  the  corresponding  log.  from  B  =  0. 08171.     Thus, 

log.  j>— B=2.64207— 0.08171=2.56036=  log.  363.4. 
The  Gauss  logarithms  would  be  applicable  in  the  solution  of  the  exponentials 
on  page  269. 

(4)  Find  by  the  Gauss  logarithms  the  log.  of  3/2004-  ^100- 

(5)  Also  the  log.  of  [(0.7345)3-f  (0.2349)3]. 

(6)  Also  the  log.  of  the  difference  ( V36—  ^27). 

(7)  Also  of  {(1.237)"— (0.9864)15}. 

219.  Let  us  resume  the  equation 

N=a*.  « 

1°.  If  rt>l,  making  a:=0,  we  have  N=l  ;  the  hypothesis  x=\  gives 
N=a.  As  x  increases  from  0  up  to  1,  and  from  1  up  to  infinity,  N  will  in- 
crease from  1  up  to  a,  and  from  a  up  to  infinity  ;  so  that  .r  being  supposed  to 
pass  through  all  intermediate  values,  according  to  the  law  of  continuity,  N  in- 
creases also,  but  with  much  greater  rapidity.     If  we  attribute  negative  values 

to  x,  we  have  N=a_x,  or  N  =— .     Here,  as  x  increases,  N  diminishes,  so 

that  x  being  supposed  to  increase  negatively,  N  will  decrease  from  1  toward 
0,  tlie  hypothesis  z=co  gives  N=0  ;  i.  c,  the  logarithm  of  zero  is  an  infinite 
negative  quantity. 

2°.  If  a<l,  put  a  — j,  where  Z>>1,  and  we  shall  then  have  N=i-,  or 

N=l',  according  as  we  attribute  positive  or  negative  values  to  x.  We  hero 
arrive  at  tho  same  conclusion  as  in  the  former  case,  with  this  difference,  that 
when  x  is  positive  N <0 .  and  when  x  is  negat ive  N  ~>  1 . 

3°.  If  a  =  l,  then  N  =  1,  whatever  may  be  the  value  of.r. 

From  this  it  appears  that, 

I.  //;  every  system  of  logarithms  the  logarithm  of  I  is  0,  and  the  logarithm 
oftlte  base  is  1. 


LOGARITHMS.  269 

II.  If  the  base  be  >1,  the  logarithms  of  numbers  >1  are  positriee,  and  the 
logarithms  of  numbers  <1  are  negative.  The  contrary  takes  place  if  the  base 
be  <1. 

III.  The  base  being  fixed,  any  number  has  only  one  real  logarithm  ;  but  the 
same  number  has  manifestly  a  different  logarithm  for  each  value  of  the  base,  so 
that  every  number  has  an  infinite  number  of  real  logarithms.  Thus,  since 
92=81  and  34=81,  2  and  4  are  the  logarithms  of  the  same  number  81,  accord- 
ing as  the  base  is  9  or  3. 

IV.  Negative  numbers  have  no  real  logarithms ;  for,  attributing  to  x  all 
values  from  — co  up  to  -\-cd  ,  we  find  that  the  corresponding  values  ofN  are 
positive  numbers  only,  from  0  up  to  -f-ao  . 

520.  In  order  to  solve  the  equation 

c=ax, 
where  c  and  a  are  given,  and  where  x  is  unknown,  wo  equate  the  logarithms 
of  the  two  members,  which  gives  us 

log.  c=x  log.  a. 
Whence 

x=  l°g'C 
"  log.  a 

To  determine  the  value  of  x  in  the  equation 

'  Aa*+Btt*-b+CaI-c+ =P, 

we  "have  * 

^A+l  +£        + >=P' 

or 

Q«*  =P, 

substituting  Q  for  the  term  in  the'  parenthesis. 

log.  P-  log.  Q 


log.  a 


If  we  have  an  equation  aL=zb,  where  z  depends  upon  an  unknown  quantity, 
x,  and  we  have 

z=A.ru4-Bxn-14- 

Since  z=  r— — =K  some  known  number,  the  problem  depends  upon  the  solu- 
tion of  the  equation  of  the  nth  degree 

KrrA^  +  B^^-f 


For  example,  let 


Hence 


'© 


:2— 5*+4 

=  9. 


/2\  9 

(.r*_5:r+4)  log.  [-)  —  log.  - 

...  x"-— 5x+4  =—2;* 

an  equation  of  the  second  degree,  from  which  we  find  z=2,  x=3. 

(3\«     9  3  9 

o)  ==7  •'•  2  '°a-  7>r—  1°S-  "J  an(l  log 

3  ,       2 

2~  °  3" 


27U  ALGEBRA. 

To  find  the  value  of  X  from  tlio  equation  > 

a 

b"    »==cmjyi-p 

1  aking  the  logarithms  of  each  member, 

("— -)  Iog-  b=mx\og.  c+(x- p)  log./, 
or 

(ro  log.  c+  log./).i-2— (n  log.  b+p  log.  f)x+a  log.  6=0, 
a  quadratic  equation,  from  which  the  value  of  x  may  be  determined. 
In  like  manner,  from  the  equation 

cmx=a&nI_1, 
we  find 

log.  a —  log.  b 
~m  log.  c — n  log.  b' 
Equations  of  this  nature  are  called  Exponential  /.' 
To  resolve  the  exponential  equation 

/117\»     8493 
V337/    =   73 
By  the  rule, 

x(log.  117—  log.  337)=  log.  8493—  log.  73 
log.  8493—  log.  73 
'  X=  ~ log.  337—  log.  llf  = 
Calculation,  • 

8493  log.  3.9290G11  337  log.  2.5276299 

73  log.  1.8633229      I      117  log.  2.0681859 

diff.  2.0657382  .  . .  .    log.  0.3150752 

diff.  .0.4594440  log.  1.6622326 

x=—  4.49616  log.=diff.  0.6528426 
This  example  admits  the  use  of  the  Gauss  logarithms. 

Let  10<  =  — 100  .'.x  log.  10=  log.  (  —  100)  ;  log.  (  —  100)  hero  must  be  re- 
garded, like  an  imaginary  quantity,  as  a  symbol  of  absurdity.  It  is  evident  that 
there  is  no  power  of  10  equal  to  — 100. 

221.  Let  N  and  N  +  l  be  two  consecutive  numbers,  the  difference  of  their 
logarithms,  taken  in  any  system,  will  be 

log.  (N  +  l)-  log.  N=  log.  (^p)=  log-  (l+^)« 

a  quantity  which  approaches  to  the  logarithm  of  1,  or  zero,  in  proportion  as 

■r}  decreases,  that  is,  as  N  increases.     Hence  it  appears  that 

The  ill (Terence  of  the  logarithms  of  two  consecutive  numbers  is  less  in  propor- 
tion as  the  numbers  themselves  are  greater. 
Let  ax  =  N  and  AV  =  N  :  then  we  have 

.r=  log.  N  to  the  base  a,  or  x=  log.  „N* 
?/=  log.  N  i"  the  base  l>.  <>r  //=  log.  i,N. 
Hence  log.  aN=  log.  ,&-"=</  log.  J>  (  Kv\.  21  1.  111.); 

.-.  r=y  log.  J>, 

*  Understanding  by  the  not  A  the  logarithm  of  N  in  the  lyatem  whoae  baaa 

bo. 


LOGARITHMS.  271 

and 

y=  , r  •  x ; (A) 

J       log.  J) 

anil  by  means  of  this  equation  we  can  puss  from  one  system  of  logs,  tc  another, 

by  multiplying  x,  the  log.  of  any  number  in  the  system  whose  base  is  a,  by  the 

reciprocal  of  log.  b  in  the  same  system ;  and  thus  we  shall  obtain  the  log.  of 

the  same  number  in  the  system  whose  base  is  b. 

The  factor  : -,   is  constant  for  all  numbers,  and  is  called  the  Modulus, 

log.a« 

that  is  to  say,  if  Ave  divide  the  logs,  of  the  same  number  c,  taken  in  two  sys- 
tems, the  quotient  will  be  invariable  for  these  systems,  whatever  may  be  the 
value  of  c,  and  will  bo  the  modulus,  the  constant  multiplier  which  reduces  the 
first  system  of  logs,  to  the  second.* 

If  we  find  it  inconvenient  to  make  use  of  a  log.  calculated  to  the  base  10,  we 
can  in  this  manner,  by  aid  of  a  set  of  tables  calculated  to  the  base  10,  discover 
the  logarithm  of  the  given  number  in  any  required  system. 

For  example,  let  it  be  required,  by  aid  of  Briggs's  tables,  to  find  the  log.  of 

2  .  5 

-  in  a  system  whose  base  is  -. 

Let  x  be  the  log.  sought,  then  by  (A) 

2 
Jbg.g 


'       5 

&  7 


log.  2  —  log.  3 

log.  5 —  log.  7' 

Taking  these  logs,  in  Briggs's  system,  and  reducing,  we  find 

—0.17609125 

1 '=  —  0.14612804 

o  5 

=  1.2050476=  log.  ^  to  base  -. 

2  3 

Similarly,  the  log.  of  -,  in  the  system  whose  base  is  5,  is 

log.  2—  log.  3 
X~log.  3— log.  2~~   ' 
which  is  manifestly  the  true  result ;    for  in  this  case  the  general  equation 

N  =a*  becomes  -=  (- )  =  ( - )     ,  and  x  is  evidently  =  —  1. 

In  a  system  whose  base  is  a,  we  have 

log.  n  , 

for,  by  the  definition  of  a  logarithm  in  the  equation  n=ax,  x  is  the  log.  n. 
Tn  like  manner, 

n,=alog.  («")=/log.« 

,*  The  term  Modulus,  of  a  system  of  logarithms,  is  generally  understood  to  be  the  num 
ber  by  which  it  is  necessary  to  multiply  Napierian  logarithms  of  numbers,  in  order  to  ob- 
tain the  logarithms  of  the  system  in  question.  The  peculiar  character  of  Napierian  loga- 
rithms will  be  presently  explained. 


272  ALGEBRA. 

\MTLES   FOR  EXERCISE. 

(1)  Given  22x+2x=12  to  find  the  value  of  x. 

(2)  Given  x-\-y  =  a,  and  m(>:_y '=«  to  find  x  and  y. 

(3)  Given  mxnx=a,  and  hx=ky  to  find  x  and  3/. 

answers. 

(1)  x=l-5849G2,  or  x=log.  (  — 4)-j-  log.  2. 

(2)  x= .',  j a+  log.  n4-  log.  »n }  and  y=  !,\a—  log. n -^-  log.  ?n } . 

(3)  x=  log.  a-^-(log.  ?«+  1°S-  7!)  ar,d  .V=7j°g'  ce-t- (log.  m  +  l°g-  ra)- 

THE   EXPONENTIAL  THEOREM. 

222.  It  is  required  to  expand  a*  in  a  series  ascending  by  the  powers  of  x. 

Since  a  =  1  -f-  a  —  1 ,  therefore  ax  =  \l-\-(a  —  1 ) } x ;  and  by  the  binomial  theorem 

we  have 

t        ,  1  x(x— 1)  x(x— l)(x— 2), 

{l  +  (a-l)^  =  l  +  x(a-l)+-4^(a-ir-+-^-T^ L(a-1)3+-- 

=  l+{(a-l)-l(a-iy-+}s(a-iy-\(a-iy+....\x+Bx* 

+  C.C3... 

where  B,  C denote  the  coefficients  of  x2,  x3 ;  and  if  we  put 

A=(a-I)-\(a-iy-+l(a-iy+l(a-iy+ 

ThenaI  =  l  +  Ax+Bx2+Cx3-r-Dx<+Ex''+ 

For  x  write  x-\-h  ;  then  we  have 

aI+h  =  l  +  A(x+/j)  +  B(x+/()-+C(r-r-/;y5-f 

=  l-fAx-f-  Bx°-+   Cx3  +   Dr»        + 

4- Ah  4-  2Bxh+3Cx-h+4'Dx3h      + 

-f  Bh-+3Cxh~  +  GDx-h-     + 

+   C/i3  +4D.*      + 

+  m*     + 

ButaJ+h=aIxa*=(l+Ax+Bx2+Cx3+....)(l  +  A/i  +  B/i2+C/i3+ 
=  1  4-Ax+Bx2   -fCx3      +D.X1      +.... 

+  A/i+A2x/i  +  ABx2/i+ ACrVi  +  . . . . 

+  }ik-    Jf-XB.rh- -{-])■  rlr  +  .  . . . 

+  l.7r      4-ACx/t3-r-.... 

4-D/t*    +  .... 

Now  these  two  expansions  must  be  identical ;  and  we  must,  therefore,  have 

tne  coefficients  of  like  powers  of  x  and  //  equal;  hence 

A* 
2B=A2         •••  B  =  — 

AB       \ 

3C  =  AB  C=— =  ;- 

AC       v 
4D  =  AC  D=-=— 

&c.    flee.  &c.  &c. 

\              \               A*X* 
Hence  ax  =  l  +  Ax+— "+ \  ....;;  +  \  ij^H 

which  is  the  exponential  thcoit  m  ;   when' 

^=(a-l)-J(a-l)«+S(a-l)  -i(a-l)H- 


LOGARITHMS.  273 

Let  e  be  the  value  of  a,  which  renders  A  =  l,  then 

(«_i)-4(e-i)»+K«-iy-K«-i)«+...=»i 

X2  3?  X* 

Now,  since  this  equation  is  true  for  every  value  of  x,  let  x=l  ;  then 
e— 1  +  1  +  1.2+1.2.3+1.2.3.4+ 

="l+l+«l)+*^)+*G^)  + 

=2-718281828459 

223.   We  add  another  method  of  calculating  the  logarithm  of  any  given 
number. 

Let  N  be  any  given  number  whose  logarithm  is  .r,  in  a  system  whose  base 
is  a ;  then 

a*  =  N  and  a"=Nx. 

Hence,  by  the  exponential  theorem,  we  have  from  the  last  equation 

1  +  A.rz+A^+---  =  1  +  A,z+A1^+....; 

and  equating  the  coefficients  of  z,  we  get  Ax=A1 ;  hence 

A,      (N-l)-i(N-l)2+!(N-l)3-.... 
X~  A  ~(a  -l)-i(a  -iy+h(a  -l)3-  . ... ; 
because  A  =(a  —  1) — \{a  — l)9+,i(a  — l)3 — ...  in  the  expansion  of  an 
and         At  =  (N— 1)— |(N  — 1)3+^(N  — l)3 in  the  expansion  ofW 

224.   To  find  the  logarithm  of  a  number  in  a  converging  series. 
We  have  seen  that  if  ax=N,  thon 

(N-l)-J(N-l)'+^(N-l)'-j(N-l)«+... 
*-(a  _l)_i(tl  -!)»+*(«  -lf-l(a  -1)*+... 
Now  the  reciprocal  of  the  denominator  is  the  modulus  of  the  system  ;*aad, 
representing  the  modulus  by  M,  we  have 

x=  log.  N  =  M{(N-l)-KN-l)2+J(N-l)3-i(N-l)*+...} 
Put  N  =  l  +  n  ;  then  N — 1=»,  and  we  have 

log.  (l4-n)=M(  +  n— in«+in3— Jn«+J»s— ...)  .  .  .[Al 

.Similarly,  log.  (1—  »)=M(-n- >s— |»s— in*— $n* ) 

...  bg.  (1+n)—  log.  (1—  tt)=2M(n+j7i3+$nB-Kn7+...) 

or  loS-  S  =2M(n+-J»3+Jn«+^n'+  ...) 

•  If,  in  the  expression  for  a*  deduced  in  (Art.  222),  we  make  z=-rt  we  obtain 

1       it       i 

which  is  tho  value  of  e,  given  at  the  end  of  the  same  art. : 
.-.  aA=£ .".  <z=:ea  /.  A  log;,  e=  log.  a  .:  —= 


A      log.  a      log.  a' 

if  t  be  the  base  cf  the  system  of  logarithms  expressed  by  log.  ♦  Therefore  — =  ia 

oy  a  previous  definition  (Art.  221),  the  modulus  for  passing  from  the  system  whose  base  ia 
to  that  whose  base  is  a.    If  log.  a  refers  to  the  base  a,  -  becomes  oqaal  to  log.  c 

A 

s 


274  ALGEBRA. 

1  I'-f -•  2P  l  +  n      I'-j-l 

Put  *=2P+7;  then  1+n== 2P+I'  1-"=2P+T'  and  I^7t=_p-: 

consequently, 

log.  (P  +  i)-  log.  P=2M  \  -^+5^-_+5pJ_+...  I 

.••log.(P+l)=log.P+2M  {  ^+3(2^+5(2^+...  | 

Hence,  if  log.  P  be  known,  the  log.  of  the  next  greater  Dumber  can  be  found 
by  this  rapidly  converging  series. 

By  substituting  the  -cries  of  Dataral  numbers  for  N  n  this  formula,  the  cor- 
responding values  of  x  will  be  their  logarithms. 

224.  To  find  the  Napierian  logarithms  of  numbers. 
In  the  preceding  series,  which  we  have  deduced  for  log.  (P  +  l).  Wl"  '""1  & 
number  M,  called  the  modulus  of  the  system  ;  and  we  must  assign  some  value 
to  this  number  before  we  can  compute  the  value  of  the  series.  Now.  as  the 
value  of  M  is  arbitrary,  we  may  follow  the  steps  of  the  celebrated  Lord 
Napier,  the  inventor  of  logarithms,  and  assign  to  M  the  simplest  possible 
value.     This  value  will  therefore  be  unity,  and  we  have 

log.  (P  +  l)=log.  P  +  2  \  .Tp^  +  3(.2p1+1)3+^p^Ty,+  •••  \ 
Expounding  P  successively  by  1,  2,  3,  4,  &c.,  we  find 

log.    3=  log.  2+2Q+3-^+^+^-7+...)  =1-0080123 
log.    4=2  log.  2 =1-3862944 

log.    5=  log.  4+2(^+3^+—^+  -)  =1^4379 
log.    6=  log.  2+  log.  3 =1.7917595 

,og.    7=  log.  6+2(^+3-^+^,+ )  =1-0459101 

log.    8=  log.  2+  log.  4,  or  3  log.  2 =2-0794415 

log.    9=2  log.  3 =2-1972246 

log.  10=  log.  2+  log.  5 =2*3025851 

In  this  manner  the  Napierian  logarithms  of  all  numbers  may  be  computed. 

225.    T<>  Ibid  the  common  logarithms  of  nunr 
The  base  of  the  Napierian  system  is  £=2-718281828...,  and  the  base  of  tbe 
common  system  is  6  =  10,  the  base  of  our  common  system  of  arithmetic  5  then 
we  have  6  =  10,  and  a=e=2»718281828...,  and  consequently,  if  N  denote  any 
number,  vre  shall  have 

log.  10N=  j— — Jq  •  log.  £N  ;  that  is, 
com.  log.  N=2.30g5851^  Nap.  log.  N=-43429448X  Nap.  I 

"  To  liml  the  value  of  trio  Napierian  base,  observe  that,  since  com.  log.  N='43499448X 
Nap   log.  N.,  if  wo  make  in  this  expression  N=c,  the  Napier! 

com.  log.  (='43499448. 
From  a  table  of  common  logs.,  therefore,  we  find  the  number  corresponding  to  the  log* 

• 


LOGARITHMS.  275 

and  the  modulus  of  the  common  system  is,  therefore, 

1 

M= r-rr:  =-43429448  .-.  2M  = -86858896 

2-3025851 

Hence,  to  construct  a  table  of  common  logarithms,  we  have 
Jog.  (P  +  l)=  log.  P  +  -86858896  \  $^±%^tf+ 5{2P\1)6+  ■   ■  \ 
Expounding  P  successively  by  1,  2,  3,  &c,  we  get 

log.    2  =  -8685889gQ+^+-^+...) 

=  •86858896  X -3465736 =  -3010300 

log.  3=  log.  2  +  .86858896Q  +  _L+l  +  ...)  .  .  =  -4771213 

log.  4=2  log.  2 =  -0020600 

log.    5=  log.  1jJ°=  log.  10—  log.  2  =  1—  log.  2    .  .   =   -6989700 
log.    6=  log.  2+  log  3 =   -7781513 

log.    7=  log.  G  + -86858896 (-+^+t^33+...)=   -8450980 

log.    8=  log.  23=3  log.  2 =   -9030900 

log.    9=  log.  32=2  log.  3 =   -9542426 

log.  10= =1-0000000 

&c.  &c. 

1-4-77. 

226.  Since  log.  — |— =2M(n+lri3+|n5+>7+ ...) 

14-71  P— 1 

let =  P  ;  then  l+n=P(l  — n),  or  w  =  p 

cp_i      i     /P— 1\3     1    /P  — IV"  I 

••^•p=2MSpqp+rlp+l)+5-lp+l)+'--i 

and  thus  we  have  a  series  for  computing  the  logs,  of  all  numbers,  without 
knowing  the  log.  of  the  previous  number. 

EXAMPLES. 

(1)  Given  the  log.  of  2  =  0-3010300,  to  find  the  logs,  of  25  and  -0125. 

100      103 
Here  25=—=—;  therefore  log.  25=2  log.  10—2  log.  2=1-3979406.    - 

125         1  1 

ASain'  '0125  =  10000=80=10^23 

.-.  log.  -0125=  log.  1—  log.  10  —  3  log.  2=— 1  — 3  log.  2=2-0969100 

(2)  Calculate  the  common  logarithm  of  17. 

Ans.  1.2304489. 

(3)  Given  the  logs,  of  2  and  3  to  find  the  logarithm  of  22-5. 

Ans.  1  +  2  log.  3—2  log.  2. 

(4)  Having  given  the  logs,  of  3  and  -21,  to  find  the  logarithm  of  83349. 

Ans.  6  +  2  log.  3+3  log.  -21. 

rithra  -43429448,  which  is  2-7182818,  the  Napierian  base.  This  also  furnishes  us  with  an- 
other definition  of  the  modulus  of  the  common  (or  any  other)  system  of  logarithms  ;  it  is  the 
common  (or,  &c.)  logarithm  of  the  Napierian  base.  See  further  note  at  the  end  of  Progres- 
sions. 


876  ALGEBRA. 


PROGRESSIONS. 

ARITHMETICAL  PROGRESSION. 

227.  When  a  series  of  quantities  continually  increase  or  decrease  by  the 
addition  or  subtraction  of  the  same  quantity,  the  quantities  are  said  to  be  'id 
Arithmetical  Progression.  A  more  appropriate  name  is  Progression  by  Dif- 
fer t.j-.es. 

Thus  the  numbers  1,  3,  5,  7, which  differ  from  each  other  by  the  ad- 
dition of  2  to  each  successive  term,  form  what  is  called  an  increasing  arith 
metical  progression,  or  progression  by  differences,  and  the  numbers  100,  97, 

94,  91, which  differ  from  each  other  by  the  subtraction  of  3  from  each 

successive  term,  form  what  is  called  a  decreasing  progression  by  differences. 

Generally,  if  a  be  the  first  term  of  an  arithmetical  progression,  and  6  the 
common  difference,  the  successive  terms  of  the  series  will  be 

a,  a±<5,  a ±2(5,  a±3<5, 

in  which  the  positive  or  negative  sign  will  be  employed,  according  as  the  series 
is  an  increasing  or  decreasing  progression. 

Since  the  coefficient  of  (5  in  the  second  term  is  1,  in  the  third  term  2,  in  the 
fourth  term  3,  and  so  on,  in  the  »tt  term  it  will  be  n— 1,  and  the  nth  term  of 
the  series  will  be  of  the  form 

a±(n-l)«J (1) 

In  what  follows  we  shall  consider  the  progression  as  an  increasing  one,  since 
all  the  results  which  we  obtain  can  bo  immediately  applied  to  a  decreasing 
geries  by  changing  the  sign  of  6. 

228.  To  find  the  sum  ofn  terms  of  a  series  in  arithmetical  progression. 

Let  a=  first  term. 
1=  last  term. 
«5=  common  difference. 
n=  number  of  terms. 
S=  sum  of  tho  series. 

rhen  S=a+(a+<5)  +  (a+2<5)+ +!. 

Write  the  same  series  in  a  reverse  order,  and  we  have 
S=         I  +(Z_«5)+(!-2d)+ +  a 

Adding,2S  =  (fl-H)  +  (a  +  0  +  (a+0  +......+(«+*) 

=n{a-\-l),  since  the  series  consists  ofn  terms. 


s=n-<^ (2) 


•  > 


Or,  since  l=a+{n  — 1)6  (Art.  227), 

sjna+««-iy (3) 

Hence,  if  any  three  of  the  five  quantities  a,  I,  (\  n,  S  be  given,  the  remain- 
ing two  may  be  found  by  eliminating  between  equations  (1)  and  (2). 

It  is  manifest  from  tho  above  process  that 

The  sum  of  any  two  terms  which  arc  equally  distant  from  the  extreme  b 
w  equal  to  the  sum  of  the  extreme  terms,  and  if  the  number  of  terms  in  the  series 
be  uneven,  the  middle  term  will  be  equal  to  one  half  the  sum  of  the  extreme  terms, 
or  of  any  two  terms  equally  distant  from  the  extreme  term.';. 


PROGRESSIONS.  277 

EXAMPLE   I. 

Required  the  sum  of  60  terms  of  an  arithmetical  series,  whose  first  term  is 
b  and  common  difference  10. 
Here  a=5,  (5=10,  n  =60 

.-.    l=a-\-{n  —  1)<5=5+59X  10=595 

(5  +  595)  X  60 
•'•  b~  2 

=  600x30  =  18000=  sum  required. 

EXAMPLE  II. 

A  body  descends  in  vacuo  through  a  space  of  16^  feet  during  the  Qrst 
second  of  its  fall,  but  in  each  succeeding  second  32^  feet  more  than  in  the  one 
immediately  preceding.  If  a  body  fall  during  the  space  of  20  seconds,  how 
many  feet  will  it  fall  in  the  last  second,  and  how  many  in  the  whole  time7 

193  386 

Here  a=~L2'        "lsF'  n=20 

193  386 

7527 

:627{  feet 


12 
(193+7527)  X  20 


S  = 


=  6^33}  feet. 

EXAMPLE    III. 

To  insert  m  arithmetical  means  between  a  and  b. 

Here  we  are  required  to  form  an  arithmetical  series  of  which  the  first  and 
last  terms,  a  and  b,  are  given,  and  the  number  of  terms  =»i  +  2;  in  older, 
then,  to  determine  the  series,  we  must  find  the  common  difference. 
Eliminating  S  by  equations  (1)  and  (2),  we  have 

2a+(n— l)(5=J+a 
/ — a 

But  here  l=b,  a=a,  n=m+2 

•.  the  required  series  will  be 

«+H5)+K^)+ +K^)+(°+^-) 

or 

b-i-ma  2i+(wi — \)a  mb-\-a 

a+  -^+r  +     m+i  + +  ^+r      +J- 

(4)  Required  the  sum  of  the  odd  numbers  1,  3,  5,  7,  9,  &c,  continued  to 
101  terms  ? 

Ans.  10201. 

(5)  How  many  strokes  do  the  clocks  of  Venice,  which  go  on  to  24  o'clock, 
strike  in  the  compass  of  a  day? 

Ans.  300 


278  ALGEBRA. 

(G)  The  first  term  of  a  decreasing  arithmetical  series  is  10,  the  common 
difference  i,  and  the  number  of  terms  21;  required  the  sum  of  the  series. 

Ans.  140. 

(7)  One  hundred  stones  being  placed  on  tho  ground  in  a  straight  line,  ;it 
the  distance  of  2  yards  from  each  other;  how  far  will  a  person  travel  uno 
shall  bring  them  one  by  one  to  a  basket  which  is  placed  2  yards  from  the  Bret 
stone  ? 

Ans.  11  mik'S  and  840  yards. 

The  relations  (1)  and  (2),  in  which  five  quantities,  a,  6,  n,  I,  S,  enter,  will 

serve  to  determine  any  two  of  these  when  the  other  three  are  given.     Thus 

thay  furnish  the  solution  of  as  many  distinct  problems  as  there  are  waj 

taking  two  quantities  from  among  five ;    and,  consequently,  the   number  of 

5-4 
problems  will  be  —or  10.     In  order  that  they  may  be  possible,  it  is  necessary 

that  the  value  of  n  should  be  not  only  real,  but  entire  and  positive.  Without 
entering  into  the  details  of  the  calculation,  we  place  below  the  solutions  of 
these  ten  problems. 

I.  Given         a,  6,  n.  (  ,  .  _      .    , 

Required        I,  S.V  =«+<— W  S-M8.+(— 1M 

III.  Given         a,  n,  I.  (         I  —  a 

a   <<$== -,  S=',n{a+l). 

Required       6,  S.  (         n  —  l  ' 

IV.  Given        6,  n,  S.  <         2S—  n(n  — 1)6        2S  +  n{n  — 1)6 
Required        a,  I.  I  2n  2n 

V.  Given       a,  n,  S.  <  _^_  j     2(S~ an) 

Required        6,  I.  \  ~~  n  ~~  '    ~"  n(n  —  1)  ' 

VI.  Given        I,  n,  S.  <  _^_,  2(nl  —  S) 

Required       a,  6.  (  n  "  n{n  —  l)  ' 

VII.  Given         a,  6,  I A  ?—g  (l  +  a){l—a+6) 

Required      n,  S.  (  6      '  2<5 

VIII.  Given        a,  I.  S.  <  _2S^  (l+a)(l— a) 

Required       n,  <J.  <  n~ a  +  V  ="  2S  —  (/+a)  ' 


IX.  Given        a,  <5,  S. 


d—la-lz  V(J— 2«/):-f  8dS 


"  2d 

Required       /,  n.  ^  =a+(„_1)(5. 


X.  Given         £,  (5,  S. 


<5+2Z±  V((,4- -0-'  — 8dS 


Required       a,  n.  f         .      ,        .. .  . 
^  v.  a  =  £ —  (n  —  l)o. 

GEOMETRICAL  PROGRESSION. 

229.  A  series  of  quantities,  in  which  each  is  derived  from  that  which  im- 
mediately precedes  it,  by  multiplication  by  a  constant  quantity,  is  called  a 
Geometrical  Progression,  or  Progression  by  Quotients. 

Thus,  the  numbers  2,  4,  8,  16,  32, in  which  each  is  derived  from  the 

preceding  by  multiplying  it  by  2,  form  what  is  called  an  increasing  geometrical 


I 


•  ■ 


I 
J 


.' 


:    -         ■ 


mtmxpty  uom  »iues  ot  the  equation  by  p, 

Sp=       ap-\-ap°-\-ap*-\- -j-ap<>-i_j_ap\ 

Subtract  the  first  from  the  second, 

S(p— l)=apn— a 

•••  S=   U         ' (1) 

p  —  1  v  ' 

Or,  since 

l=apn~1 
L,     pi — a 

s=^r <2> 

If  the  series  be  a  decreasing  one,  and  consequently  p  fractional,  it  will  be 
convenient  to  change  the  signs  of  both  numerator  and  denominator  in  the  above 
expressions,  which  then  become 

S_a(l-Pn) 
1—P 
a — pi 
l—  p 

231.  If  two  progressions  have  different  first  terms,  but  the  same  ratio,  the 
ratio  of  the  sums  of  the  two  is  equal  to  the  ratio  of  their  first  terms.     For 

(a  +  ap+ap°-+ap*+,  &c.)  :   (J+Sp+^+ft/^-f ,  &c.) 
=a(l+p+  P'+  ps+,  &c.):6(l+P+  p-+  P3+,  &c.)=a:6 

232.  It  appears  that  if  any  three  of  the  five  quantities,  a,  I,  p,  n,  S,  bo 
given,  the  remaining  two  may  be  found  by  eliminating  between  equations  (1) 
and  (2).  It  must  be  remarked,  however,  that  when  it  is  required  to  find  pfrom 
a,  n,  S  given,  or  from  n,  I,  S  given,  we  shall  obtain  p  in  an  equation  of  the  n01 
degree,  a  general  solution  of  which  can  not  be  given.  If  n  be  required,  it  will  be 
convenient  to  apply  logarithms,  as  the  equation  to  be  resolved  will  be  an  expo 
nential. 


M 


2: 

: 
I 


th 

si 

8t 

' 

K 

t\ 

tn 

P; 

tl 

e 

tl 

' 


Required       a,  b.  ( 

III.  Given         a,  n,  I.  <         I —a  i_f-j_n 
Required       6,S.l        n  —  V          -   ^   t  ; 

IV.  Given        6,  n,  S.  <        8S—  n(n-l)d  ^2S+n(n-l)d 
Required        a,  I.  i  °~  2n  2n 

V.  Given        a,  n,  S.  5  .      2S  2(S-an) 

Required        6,  I.  (         n  11(71  — 1) 

VI.  Given        I,  n,  S.  <  fl_^_;  j_2(»*— S) 
Required       a,  6.1       ~  n  '  n(n  —  1) 

VII.  Given         a,  (5,  I.  <        E— a  (J  +  a)(i  — a  +  J) 

Required      n,  S.  i  "~    «J    +  '  2<» 

VIII.  Given        a,  J.  S.  <        _2S  (E+a)(i-a) 

Required       n,  J.  i  n_a  +  Z'         2S— (J+a)  " 


IX.  Given        a,  <*,  S.  )n=-  — r^j 


Required        i,  n.  ^  _a+(n_1)(i. 


X.  Given         I,  «J,  S.  >  »= r^j 

Required       a,  n.  (  a  =  i_(n_lyi 

GEOMETRICAL  PROGRESSION. 

229.  A  series  of  quantities,  in  which  each  is  derived  from  that  which  im- 
mediately precedes  it,  by  multiplication  by  a  constant  quantity,  is  called  « 
Qeometrica.1  Progression,  or  Pre  ■>  by  Quotients. 

Tims,  the  numbers  2,  4,  8,  16,  32 in  which  each  is  derived  from  the 

pm-rdim;  bv  multiplying  it  by  2,  form  what  lb  called  an  increasing  geom 


PROGRESSIONS.  279 

progression  ;  and  the  numbers  243,  81,  27,  9,  3,  ...  in  which  each  is  derived 
from  the  preceding  by  multiplying  it  by  the  number  -,  form  what  is  called  a 

decreasing  geometrical  progression. 

The  common  multiplier  in  a  geometrical  progression  is  called  the  common 
ratio. 

Generally,  if  a  be  the  first  term  and  p  the  common  ratio,  the  successive 
terms  of  the  series  will  be  of  the  form 

a,  ap,  op2,  ap2 

The  exponent  of  p  in  the  second  term  is  1,  in  the  third  term  is  2,  in  the 
fourth  term  3,  and  so  on  ;  hence  the  na  term  of  a  series  will  be  of  the  form, 

ap"-1. 

230.  To  find  the  sum  ofn  terms  of  a  series  in  geometrical  progression. 

Let  a==  first  term, 
1=  last  term, 
p=  common  ratio, 
n=  number  of  terms, 
S  =  sum  of  the  series. 
Then 

S  =a4-ap+ap2+ap34- +  crpn-1. 

Multiply  both  sides  of  the  equation  by  p, 

Sp=        ap-\-ap"-{-ap3-\- ^_ap"-i_j_ap°. 

Subtract  the  first  from  the  second, 

S(p—l)=ap"—a 
a(pn  —  l) 

Or,  since 

l=aPn~l 

*£r « 

If  the  series  be  a  decreasing  one,  and  consequently  p  fractional,  it  will  be 
convenient  to  change  the  signs  of  botli  numerator  and  denominator  in  the  above 
expressions,  which  then  become 

S__«(l-A?) 

l— P 

1—  p 

231.  If  two  progressions  have  different  first  terms,  but  the  same  ratio,  the 
ratio  of  the  sums  of  the  two  is  equal  to  the  ratio  of  their  first  terms.     For 

(a+ap+ap'i+ap*+,  &c.)  :   (b  +  bP  +  bp*  +  bp3+,  Sec.) 
=  a(l+p+   p*+  p3+,&c.):i(l+P+  p2+  p3+,  &zc.)—a:b 

232.  It  appears  that  if  any  three  of  the  five  quantities,  a,  I,  p,  n,  S,  b» 
given,  the  remaining  two  may  be  found  by  eliminating  between  equations  (1) 
and  (2).  It  must  be  remarked,  however,  that  when  it  is  required  to  find  pfrom 
a,  n,  S  given,  or  from  n,  I,  S  given,  we  shall  obtain  p  in  an  equation  of  the  71th 
degree,  a  general  solution  of  which  can  not  be  given.  If  n  be  required,  it  will  be 
convenient  to  apply  logarithms,  as  the  equation  to  be  resolved  will  be  an  expo 
nential. 


5280  ALGEBRA. 

EXAMPLE   I. 

Requirea  the  sum  of  10  terms  of  the  series  1,  2,  4,  8, . 
Here  a  =  l,  p=2,  n=zlO 

fl(p"-l) 
p  — 1 

=  OI0 J 

=  1023. 
EXAMPLE  II. 

Required  the  sum  of  10  terms  of  the  series  1,  -,  -,  — , 


3'  9'  27' 


Here  assl,  p=y  n=\0 


...S: 


2 
:3' 

q(i—  p") 
1-p 

l- 


■©' 


174075 
: 59049  ' 


EXAMPLE  III. 

To  insert  m  geometric  means  between  a  and  b.  • 

Here  we  are  required  to  form  a  geometric  series,  of  which  the  first  and  last 
terms,  a  and  o,  are  given,  and  tho  number  of  terms  =?n-\-2 ;  in  order,  then* 
to  determine  the  series,  we  must  find  the  common  ratio. 

Eliminating  S  by  equations  (1)  and  (2), 

apn~—a=pl — a 

But  here 

l  =  b,  71=771-4-2 


Henco  the  series  required  will  be 

lb  16*  lhm~l  lbm  #">+» 


or 


CL+  ™+yamb-\.  ™+fam-lb-  +  ...-{-  m+^/a'1bm-,-\-  ^ab^+b, 
or 

m  1  m— 1       1  8        m— 1  t  m 

a  -\.  a™+lb™+i + a™+1  b™+1  + . . .  +  an,+ '  6","+7 + a*"+i  £"+»  -f  b. 

233.  To  ^"fi  ^c  sum  of  an  infinite  series  decreasing  in  geometrical  jrro- 
gression. 

Wo  have  already  found  that  tho  sum  of  ii  terms  of  a  decreasing  geometric* 
series  is 


1-P* 

which  may  dc  put.  under  the  form 


PROGRESSIONS.  281 

a—apn 


S  =  r^ 


'1-P     l-p-r- 

Since  p  is  a  fraction,  pn  is  less  than  unity,  and  the  greater  the  number  n,  the 

smaller  will  be  the  quantity  pn ;  if,  therefore,  we  take  a  veiy  great  number  of 

apa 
terms  of  a  decreasing  series,  the  quantity  pn,  and,  consequently,  the  term  , 

a 
will  be  very  small  in  comparison  with ;  and  if  we  take  n  greater  than  any 

assignable  number,  or  make  n  =  co,  then  p1"1  will  be  smaller  than  any  assignable 
number,  and  therefore  may  be  considered  =0,  and  the  second  term  in  the 
above  expression  will  vanish. 

Hence  we  may  conclude  that  the  sum  of  an  infinite  series,  decreasing  in 
geometrical  progression,  is  * 

1—  P 

a 
Strictly  speaking, is  the  limit  to  which  the  sum  of  any  number  of 

terms  approaches,  and  the  above  expression  will  approach  more  or  less  nearly 
to  perfect  accuracy,  according  as  the  number  of  terms  is  greater  or  smaller 
Thus,  let  it  be  required  to  find  the  sum  of  the  infinite  series 

1 
Here  a  =  l,  p=^,  n=ao 


1-P 
1 


1 
X~3 


3 

~~2' 

3 

The  error  which  we  should  commit  in  taking  -  for  the  sum  of  the  first  n 


terms  of  the  above  series  is  determined  by  the  quantity 

apn  _3/l\n 
l^p=2\3/  ' 
™        ..  3/l\s        1  1 

Thus,  if  »=5,  tl  on  -2{~)  =27F=162; 

n  =  G,then-y  =— -— . 

3 
Hence,  if  we  take  -  as  the  sum  of  5  terms  of  the  above  series,  the  amount 

would  be  toe  great  by  y^-. 


282  ALGEBRA. 

3  1 

If  we  take  -  as  the  sum  of  G  terms,  the  amount  w.ll  be  too  great  by  —— . 

~  4  DO 

and  so  on.* 

*  I.  The  theory  of  progressions  involves  that  of  logarithms.  Let  there  be  two  progres- 
sions, the  one  geometric,  beginning  with  1,  the  other  arithmetical,  beginning  with  0. 

-ffl:2:4:8:16:32:64:128,  &c. 
-H). 3. 6. 9. 12. 15. 18.    21,  &.C., 
which  exhibit  a  notation  sometimes  employed. 

If  we  compare  these  with  each  other,  we  perceive  that,  multiplying  together  any  two 
terms  of  the  first,  and  adding  the  corresponding  terms  of  the  second,  we  obtain  two  I 
sponding  terms,  again,  of  these  same  progressions.  Thus,  4X1G=64,  6-}-12=13  ;  ai. 
perceive  that  18  corresponds  to  04.  Thus  a  multiplication  is  effected  by  addition.  Tiiid 
simple  observation  is,  no  doubt,  very  ancient ;  but  it  was  the  genius  of  Napier,  a  Scottish 
baronet,  which  derived  from  it  the  theory  of  logarithms,  one  of  the  most  useful  of  modern  dis- 
coveries.    It  was  published  in  1G44,  under  the  title  of  Mirifici  Logarithmorum  Dejcriptio. 

Logarithms,  then,  according  to  Napier,  were  regarded  as  a  series  of  numbers  in  arith- 
metical  progression,  while  the  numbers  themselves  corresponding,  formed  a  geometrical 
progression.     I  proceed  to  explain  his  method  of  constructing  them. 

In  order  that  the  geometrical  progression fluiull  embrace  all  numbers  greater  than  1,  it 
is  necessary  to  conceive  it  formed  of  tonus  which  increase  in  an  insensible  mam 
out  lVom  1 ;  and,  to  have  their  logarithms,  it  is  necessary  to  conceive  the  arithmetical  pro 
gression  as  composed  of  terms  which  vary  by  insensible  decrees,  setting  out  from  zero. 

At  their  origin,  the  simultaneous  increments  which  the  terms  1  and  0  receive  are  inap- 
preciably small ;  but,  however  small  they  may  be,  we  may  conceive  that  there  is  a  certain 
relation  established  between  them,  which  is  entirely  arbitrary.  Thus,  when  these  incre- 
ments begin  to  arise,  we  can  suppose  that  that  of  the  logarithm  0  is  double,  triple,  &c.,  of 
that  of  the  number  1.  This  relation  is  called  the  modulus  of  the  logarithms,  which  i 
nate  by  M. 

Suppose,  now,  that  to  the  terni  1  of  the  geometric  progression  an  increment  u,  1  i  ry 
small,  but  yet  appreciable  in  numbers,  is  given.  The  corresponding  increment  of  the  tenii 
zero  of  the  arithmetical  progression  will  be  very  nearly  equal  to  Mu) ;  and  we  can  take  for 
the  two  progressions  these  : 

-ff-1 : 1-f-w:  (l+w)2:  (l-fw)3:  (l+o)4:&c. 
-^-0.    Mu.   2Mu    .    3Mu    .    4Mo    .tic. 

We  have  said  that  the  relation  or  modulus  M  can  be  taken  at  pleasure  ;  consequently 
according  to  the  values  attributed  to  it,  will  be  obtained  different  systems  of  logarithms. 
The  logarithms  which  Napier  published  were  derived  from  the  progressions 

4f  1 : 1+u :  (1+up :  (1+^)3 :  &c. 
-i-0.        u.       2w    .       3cj    .&.C., 
which  supports  M=l. 

This  avoids  the  multiplications  by  M.  The  logarithms  of  numbers  in  Napier's  table 
serve  to  find  those  of  any  other  system,  by  simply  multiplying  each  by  the  modulus  of  that 
system. 

The  terms  of  these  two  series  vary  slowly,  so  that,  in  prolonging  both  as  far  as  we  please, 
we  are  sure  of  finding  in  the  first,  terms  equal  numbers  2,  3,  &c.,  or  so  nea» 

them  that  the  difference  may  be  neglected.    The  corresponding  tonus  of  the  second  may 
tfieu  be  taken  for  the  logarithms  of  these  numbers,  those  written  in  the  tables. 

By  this  we  perceive  that  these  logarithms  are  not  exactly  those  of  the  numbers  beside 
which  they  are  written.  But  there  is  another  cause  of  inaccuracy,  viz.,  that  <j  represents 
only  approximately  the  increment,  which  the  logarithm  0  takes  when  cj  is  that  taken  by  1 
The  .  i  is,  however,  the  greater  the  ezactni 

1 1  Let  it  be  proposed  to  determine  the  error  produced  by  assuming  that  the  differen  t 
the  numbers  is  proportional  to  the  difference  of  their  logarithms,  when  the  number  of  phvu'i 
in  the  numbers  is  5,  and  their  difference  not  greater  than  1. 

If  in  the  series  [A],  Art.  221,  we  make  «=-,  we  have 

'C-?)=»+"-"=Mi;-,1.+,!-s+'*°i' 


PROGRESSIONS.  283 

As  in  arithmetical  progressions,  all  the  questions  which  can  be  p-oposed  foi 

•olution  in  geometric  progressions  reduce  to  10,  the  solutions  of  which  are  de 

duced  from 

l=apn~l     (1) 

_     pi — a 

S=- (2) 

p  —  1  v   ' 

from  which  it  appears  generally  that  as  the  number  x  increases,  the  difference  of  the  loga- 
rithms of  x  and  \-\-x  diminishes.    Also,  since  -  is 

x 

diminished  by  more  than  it  is  increased,  we  have 


rithms  of  x  and  \-\-x  diminishes.    Also,  since  -  is  greater  than  the  whole  series,  -  being 

x  x 


M 
Z(l-j-x) — Ix-^—. 
x 

If  the  base  be  10,  we  have  seen  that  M=0.4342...<-.    Hence,  in  this  case, 


l(l+x)-lx<±. 


• 


If  x  consist  of  five  places,  its  least  value  is  10000.    Therefore  the  greatest  value  of 

l(l4-x) — Ix  is  less  than =0.00005. 

20000 

Hence  we  may  infer  that  the  logarithms  of  every  two  consecutive  whole  numbers  con- 
sisting of  five  places  must  agree  in  the  first  four  decimal  places  at  least. 
Now  let 

A  =  lll4-x)—lx=l1-^. 
x 

A/=Z(2+x)_Z(1+a;)=Z?±?. 
i-\-x 

A-A'=z!±f-Z2+* 


But  by  [A],  Art.  224. 


l+x 

(1+s)'       /  1     \ 

x{2+x)       \  nrx{2+x)t 


ZKl2^,)=M^ 


1 

x(2+x)!~  "  dx(2-f<r)      &F>(2+a:)a  '  3a^(2-f-a;)3 

1 


A-A'<- 


"2x(2+2;) 
If  x  consist  of  five  places,  its  least  value  is  10000,  and,  therefore,  the  greatest  valae  of 

A — A'  is  less  than = ,  which,  when  reduced  to  a  decimal,  has  no 

20000  X 10002     200040000 

significant  figure  within  the  first  eight  places.     Hence,  in  tables  which  extend  only  to 

seven  places,  we  may  assume  that  A — A'=0,  or  A=A'. 

Thus  we  infer  that,  under  the  circumstances  which  have  been  supposed,  the  logarithms 

of  numbers  in  arithmetical  progression  will  themselves  be  in  arithmetical  progression 

P 

Let  now  n  and  w-f-1  be  two  consecutive  whole  numbers,  and  n-f-—  an  intermediate  frac- 

1 
tion.    These  may  be  looked  upon  as  three  terms  of  an  arithmetical  progression,  whose  first 

1  p 

term  is  n,  whose  common  difference  is  -,  whose  (p-\-l)til  term  is  n-\ — ,  and  whose  fe-f-l}* 

term  is  n-\-\.    By  what  has  been  already  shown,  the  logarithms  of  the  several  terms  of 
this  scries  will  also  be  in  arithmetical  progression. 

Let  <5  be  their  common  difference.     The  (p+l)th  term  of  this  series  will  be 

ln-\-p8, 
which  will  be  the  logarithm  of  the  (^-f-l)th  term  of  the  former  series  ; 


.-.  ln+pS=l '«+-) [Bl 


284  ALGEBRA. 

rF]>*<i«k  solutions  are  contained  in  the  following  table  : 

I.  <?«ven         a,  p,  n.  (     _     n_,  pi— a     a(pn  —  1) 

Required       I,  S.  2  l  ~ ap"    '  ,,  — 1  ~~    p  — 1    ' 


Given  J,  p,  n.  <  I  /(p"_i) 

Required      a,  S.  I  a~^  S=p— i(p_  l) 


n— 1/7"         ■>— I; 


III.  Given         a,  w,  Z.  <         n_,/7  V*"—  V"n 
Required      p,  S.  J  p  =   V  a'  S=  ""^7—  ""-1/a  ' 

IV.  Given        p,  n,  S.  <         S(p— 1)         Spn-J(p  — 1) 

Required       a,  /.  C        "   pn — 1   '    "I        p"  —  l       ' 

V.  Given        a,  n,  S.  S  S 

Required        p,  I.  fTM^       .  +  !=-,  l=ap^. 

VI.  Given         £,  n,  S. 
Required        p,  a. 


/ 


VII.  Given         a,p,l.<pl—a  \og.l—  log.  a 

Required      n,  S.  t       -p — 1'      ~     '         log.  p 

VIII.  Given        a,  I,  S.  <        S— a      _       log.  I—  log.  a 

Required       p,n.(         S — V      ~   ~*~        log.  p 

IX.  Given        a,  p,  S.  <        a-f-S(p— 1)  log.  Z—  log.  a 

liequired        I,  n.  (  p  ■  log.  p 

X.  Given         Z,  p,  S.  (         ,         •  log.  I —  los:.  a 

t,       .     ,  <a=lp— S(p— 1),  n  =  l-\ — ^—r     -2— . 

Required      a,  rc.  c  v       /»  ~r        j0g>  ^ 

HARMONICAL  PROGRESSION. 

234.  A  series  of  quantities  is  called  a  harmonical  progression  when,  if  aut 
three  consecutive  terms  be  taken,  the  first  is  to  the  third  as  the  difference  of 
the  first  and  second  to  the  difference  of  the  second  and  third. 

Thus,  if  a,  b,  c,  d bo  a  series  of  quantities  in  harmonical  progression, 

we  shall  have 

a:c::a  —  b:b — c;  b:d::b — c:c — d,  cce. 

The  reciprocals  of  a  scries  of  terms  in  harmonical  progression  are  in  arith- 
metical progression. 

Let  a,  b,  c,  d,  c,f be  a  series  in  harmonical  progression. 

Then,  by  definition, 

Also,  the  last  term  of  the  latter  series,  which  will  be 

ln-\-qd, 
will  be  the  logarithm  of  the  last  term  of  the  former  series  ; 

.-.  l{n+l)—ln-\- q6,  .:  I{n+i)—ln=q6. 
Hut  by  [B],  lL+£\-4n 

\        ql  /(/j-j-1)  —  hi      q 

But,  also, 

(»+l )  —  n     7 
Hence  Vac  difTeroncci  of  the  logarithms  arc  as  the  diflVroncos  of  the  numbem 


-  / 


a 


285 


V 


•  / 


(a 


ca. 


in 


<y 
- 


v> 


■: 


Z'     / 


- 


i«a 


Cr    CL 


' 


eu 


SIMPLE  INTEREST. 

Problem  I. — To  find  the  interest  of  a  sum  pfor  t  years  at  the  rate  r. 

Since  the  interest  of  one  dollar  for  one  year  is  r,  the  interest  of  p  dollars  for 
one  year  must  be  p  times  as  much,  or  pr  ;  and  for  t  years  t  times  as  much  as 
for  one  year;  consequently, 

i=ptr    ....  (1) 


?84 
1 


\ 
V 

I 


23' 
three 
the  fi 

Th 
we  si 

Th 
meticc 

Le 
Then 

Alsc 
will  be 
Hut  by  [B], 

Bat,  also, 


iL+'A-i,, 


(«+i )  — »  7 

Hence  t\ic  difference!  of  the  logarithms  are  as  the  differences  of  the  numbem 


INTEREST  AND  ANNUITIES.  095 

a  c::a — b:b — c  ;  b:d::l  —  c:c — d;  c:e::c — d:d — e,  &c. 
.••  ab — ac=ac — be,  be  —  bd=bd — dc,  cd — ce=ce — cd,  &c. 
ab      ac       ac       be     1-.       hd      bd       dc     cd       ce       ce       ea 
'  abc     abc     abc     abc1  bed     bed     bed     bed'  cde     cde     cde     cie 
or 

1     11     1   1     11     1   1     11     1 
c      b     b     a'  d     c     c      V  e     d     d     c' 

from  which  it  appears  that  tho  quantities  -,  j,  -,  -?,  -,  &c,  are  in  arithtreVica. 

CI     U     C     Co     C 

progression. 

To  insert  m  harmonic  means  between  a  and  b. 

Since  tho  reciprocals  of  quantities  in  harmonical  progression  are  in  aritn 

1       ,  1 
metical  progression,  let  us  insert  m  arithmetic  means  between  -  and  r 

Generally,  in  arithmetical  progression, 

l=a+(n  — 1)6 

I — a 

.'.  (5= -. 

n  —  1 

r     ,  •  ,     1  1  ,  o—  b 

In  this  case,  I =7,  a=-,  n=m4-2,  and  .-.  ()■=-. — ,  ,,    ,. 
0         a  (m-j-l)ao 

The  arithmetic  series  will  be 

1        a+mb        2a  +  {m—l)b  (vi  —  l)a+2b        ma+b        1 

ar '(m+l)a6"'      (m-\-l)ab    + (m+l)aZ»    "^(m+l)a6"^"6* 

Therefore  the  harmonical  series  will  be 

(m-\-l)ab        (m-\-l)ab  (7?i-\-l)ab         (m-\-l)ab 

°^     a+mb    +2a+(wi— 1)6"^" "*"(m— l)a+26"*"    ma+b   + 


INTEREST  AND  ANNUITIES. 


235.  The  solution  of  all  questions  connected  with  interest  and  annuitf-M 
may  be  greatly  facilitated  by  the  employment  of  the  algebraical  formulas. 
In  treating  of  this  subject  we  may  employ  the  following  notation  : 
Letp  dollars  denote  the  principal, 
r  the  interest  of  $1  for  one  year. 
t  the  interest  of  p  dollars  for  t  years. 
s  the  amount  of  p  dollars  for  t  years  at  the  rate  of  interest  denoted 

by  r. 
t  the  number  of  years  thatp  is  put  out  at  interest. 

SIMPLE  INTEREST. 

.     Problem  I. — To  find  the  interest  of  a  sum  pfor  t  years  at  the  rate  r. 

Since  the  interest  of  one  dollar  for  one  year  is  r,  the  interest  of  p  dollars  for 
one  year  must  be  p  times  as  much,  or  pr  ;  and  for  t  years  t  times  as  much  as 
for  one  year;  consequently, 

i=ptr    ....  (1) 


286  ALGEBRA. 

Tuoklkm  II. —  To  find  the  amount  of  a  sum  p  laid  out  for  t  years  at  simple 
in  '■■  rest  at  (he  rate  r. 

The  amount  must  evidently  be  equal  to  the  principal,  together  with  the  in- 
terest upon  that  principal  for  the  given  time. 

Hence  s=p+ptr 

=p(l  +  tr) (2) 

EXAMPLE   I. 

Required  the  interest  of  $873.75  for  2\  years  at  4  J-  per  cent,  per  annum. 
It  will  be  convenient  to  reduce  broken  periods  of  time  to  decimals  of  a  year. 
By  the  formula  (1)  wo  have 

i=ptr. 
In  tho  example  before  us, 

p  =$873.75 

r  =  $.0475* 

I  =21  years =2.5  years. 

.-.  z"=873.75x  2.5  X -0475  dollars. 
=$103.7578125. 
The  amount  of  tho  above  sum  at  tho  end  of  the  given  time  will  be 

s=p-\-ptr 
=  $873.75  +  5103.757. 

PRESENT    VALUE  AND   DISCOUNT   AT   SIMPLE  INTEREST. 

The  present  value  of  any  sum  s  due  t  years  hence  is  the  principal  which  in 
the  time  t  toill  amount  to  s. 

The  discount  upon  any  sum  due  t  years  hence  is  the  difference  between  that 
sum  and  its  present  value. 

Problem  III. —  To  find  the  present  value  of  s  dollars  due  t  years  hence, 
simple  interest  being  calculated  at  the  rate  r. 

By  formula  (2)  we  find  the  amount  of  a  sump  at  the  ond  of  t  years  to  be 

s=j)-\-plr. 
Consequently,  p  will  represent  the  present  value  of  the  sum  s  due  t  years 
hence,  and  we  shall  have 

^njh-r <3> 

for  the  expression  required. 

*  r  is  the  interest  of  Si  for  one  year.    To  find  tho  value  of  r  when  interest  is  calculated 

at  the  rate  of  $4j  or  $4.75  per  cent,  per  annum,  we  have  the  following  proportion : 

$100  :Sl::  $4.73  :r 

•  4.7."i 

.\r=$ =$0.0475. 

100 

In  like  manner, 

When  the  rate  of  interest  per  cent,  is  $7,    then  r=$0.07. 

When  the  rate  of  interest  per  cent,  is  6,    then  r=   0.06. 

When  the  rate  of  interest  per  cent  is  5,    then  r=   0.05. 

When  lli.'  rate  of  interest  per  cent,  is  4],  then  r=   0.0475. 

When  the  rate  of  interest  per  cent,  is  4*,  then  r=  0.045. 

When  the  rate  of  interest  per  cent,  is  4|,  then  r==   0.0 

When  the  rate  of  interest  per  cent,  is    4.   then  r—  o.oi. 
When  the  rate  of  interest  per  cent,  is     '1\,  thenr=   0.0375. 
dec.  4.c. 


INTEREST  AND  ANNUITIES.  287 

P.wquired  tho  present  vame  of  100  dollars,  payable  in  10  years,  at  7  per  cent, 
per  annum. 

In  this  example  s  =  100 

tz=  10 
r  =  .07 

100 
•••  *=1  +  10X.07=S58-82- 

Problem  IV. —  To  find  the  discount  on  s  dollars  due  t  years  hence,  at  the 
rate  r,  simple  interest. 

Since  the  discount  on  s  is  the  difference  between  s  and  its  present  value,  we 
shall  have 

s 


d=s 


1  +  tr 
str 

(4) 


~~  l  +  /r 

EXAMPLE. 

Required  the  discount  on  $100,  due  3  months  hence,  interest  being  calcu- 
lated nt  the  rate  of  5  per  cent,  per  annum. 
Here  s  =$100 

£=3  months  =    .25  years. 
r—  =8.05. 

100X.25X-05 
'"     ~  l-f-.25X.05 

1.25 


— 1.0125 
=$1,235  dis. 

ANNUITIES   AT   SIMPLE  INTEREST. 

Problem  V. —  To  find  the  amount  which  must  he  paid  at  the  end  oft  years, 
for  the  enjoyment  of  an  annuity  a,  simple  interest  being  allowed  at  the  rate  r. 
At  the  end  of  the  first  year  the  annuity  a  will  be  due  ;  at  the  end  of  the 
second  year  a  second  payment  a  will  become  due,  together  with  ar  the  in- 
terest for  one  year  upon  the  first  payment ;  at  the  end  of  the  third  year  a 
third  payment  a  becomes  due,  together  witu  2ar  the  interest  for  one  year 
upon  the  former  two  payments,  and  so  on ;  the  sum  of  all  these  will  be  the 
amount  required. 
Thus: 

At  the  end  of  the  first  year,  the  sum  due  is      a. 
At  the  end  of  the  second  year  the  sum  due  is  a-{-ar. 
At  the  end  of  tho  third  year,  the  sum  cue  i?      a4-2ar. 
At  the  end  of  the  fourth  year,  the  sum  due  is  a-\-3ar 

&c.  &c.  &c. 

At  the  end  of  the  tth  year,  the  sum  due  is  #+(f — l)ar 

Hence,  adding  these  all  together  for  the  whole  amount, 

5  =  to-L.ar(l  +  2+3+.  .  .  .      (<— 1))- 
Or,  taking  the  expression  for  the  sum  of  the  arithmetical  senes,  1-J-2-J-3 

4- («-l) 

t(t—l) 
s=ta+ra.    ±  g      (5) 


288  ALGEBRA. 

Problem  VI. — To  find  the  present  value  of  an  annuity  npayablefor  t  years 
simple  interest  being  allowed  at  the  rate  r. 

It  is  manifest  that  the  present  value  of  the  annuity  must  be  a  sum  such  that, 
if  put  out  at  interest  for  t  years  at  the  rate  r,  its  amount  at  the  end  of  that 
period  will  be  the  same  with  the  amount  of  the  annuity. 

Hence,  if  we  call  this  present  value  p,  we  shall  have,  by  Problems  I.  and  V., 
p-^-ptr=  amount  of  annuity. 
t{t—l) 

t(t—l) 
ta-\-ra.- 


p=      '  1.2 

1  +  tr 
_ta  2+(<— l)r 
~~2  '  ~~ : l  +  tr      ' 


(6)' 


COMPOUND  INTEREST. 

Problem  VII. —  To  find  Oie  amount  of  a  sum  p  laid  out  for  t  years,  com- 
pound interest  being  allowed  at  the  rate  r. 

At  the  end  of  the  first  year  the  amount  will  be,  by  Problem  II., 

p+pr,  orp{l-{-r). 

Since  compound  interest  is  allowed,  this  sum  ^(l+r)  now  becomes  the 
principal,  and  hence,  at  the  end  of  the  second  year,  the  amount  will  be 
p(l-\-r),  together  with  the  interest  onp(l-\-r)  for  ene  year;  that  is,  it  will  be 

P(l+r)+pr£+r)>  orjp(l+r)«. 

The  sum  p(l+r)5  must  now  be  considered  as  the  principal,  and  hence  the 
whole  amount,  at  the  end  of  the  third  year,  will  be 

p(l+ry+pr(l  +  r)-,  or^l  +  r)3. 
And,  in  like  manner,  at  the  end  of  the  tih  year,  wo  shall  have 

s=p(l  +  ry (7) 

Any  three  of  the  four  quantities,  5,  p,  r,  t,  being  given,  the  fourth  may  al- 
ways be  found  from  the  above  equation. 

EXAMPLE  I. 

Find  the  amount  of  $15.50  for  9  years,  compound  interest  being  allowed 
at  the  rate  of  3,V  per  cent,  per  annum,  the  interest  payable  at  the  end  of 
nacli  year. 

By  equation  (7), 

s=p(l  +  ry 
.-.  log.  s=  log.  p+t  log.  (1-f-r). 
But  J9=S15.50 

t  =  9  years 
r  =  S.035 
.-.  log  ^  =  1.1903317 
Hog.  (l+r)  =  0. 1344627 

.-.log.  5  =  1.3247911  =  log.  of  21.12181 
.-.  8=321.12481. 

*  It  is  unnecessary  to  give  any  examples  under  this  rule,  as  the  purchase  of  annuities 
nt  simple  interest  can  never  be  ol 


INTEREST  AND  ANNUITIES.  289 

EXAMPLE  II. 

Find  the  amount  of  .£182  12s.  6cl.  for  18  years,  6  months,  and  .  0  days,  at 
the  rate  of  3£  per  cent,  per  annum,  compound  interest,  the  interest  being 
payable  at  the  end  of  each  year. 

In  this  case,  it  will  bo  convenient,  first,  to  find  the  amount  at  compound  in- 
terest of  the  above  sum  for  18  years,  and  then  calculate  the  interest  on  the 
result  for  the  remaining  period. 
By  formula  (7), 

s=P(l  +  ry 
log.  s=log.  p+t  log.  (1+r) 

Here         p=<£l82.  12s.  6d.=  £182. 625 
r=  =£.035 

t=  =18  years 

.-.  log.  p =2.2615602 
Hog.  (1+r)  =  0.2689254 

.-.  log.  s  =  2.5304856=  log.  of  339.224. 
Again,  to  find  the  interest  on  this  sum  for  the  short  period,  we  have 

i=st'r 
.'.  log.  i=  log.  s-\-  log.  i'-|-  log  r. 
Here    5=66339.224 
r=c£.035 

t'=6  months,  10  days—     .527402  years 
.-.  log.  s  =  2.5304856 
log.  r=2.5440680 
log.  i' =1.7221401 


.-.  log.  si'r=0.7966937=  log.  of  6.2617200 

.-.  srr=£6.26172. 

J'he  whole  amount  required  will,  therefore,  be 

s+s  V  r=c£339.224+c£6.26172 

=£345  95.  8±d. 
I  2 

EXAMPLE  III. 

Required  the  compound  interest  upon  $410  for  2|  years  at  4|  per  cent,  per 

annum,  the  interest  being  payable  half  yearly. 

In  this  case  the  time  t  must  be  calculated  in  half  years ;  and,  since  we  have 

r 
supposed  r  to  be  the  interest  of  $1  for  one  year,  we  must  substitute  -,  which 

will  be  the  interest  of  $1  for  half  a  year  ;  the  formula  (7)  will  thus  become 

S=i,(1+l)31 


.-.  log.  s=  \og.p+ 21  log.  (l  +  o)  ■ 


Here  p  =  $410 

r=$.045 


21 =5  half  years 

.-.log.;;  =  2.6127839 
5  log.  1.0225  =  0.0483165 


log.5=2.66110()4  =  log.  of  456.2471 
.••s=$458.2471. 
T 


£90  ALGEBRA. 

The  inte    »    ~*iust  t>cs  the  difference  between  this  amount  ar  \  the  original 
principal ; 

.'.i=s — p 

=  $458.247— $410 
=§48.247. 

EXAMPLE  IV. 

$400  was  put  out  at  compound  interest,  and  at  the  end  of  9  years  amounted 
to  $569,333  ;  required  the  rate  of  interest  per  cent. 
Here  s,  p,  t  are  given,  and  r  is  sought. 
From  formula 

we  have  log.  (1 +  ?■)=-(  log.  s — log.^>). 

Here  s  =  8569.3333 

;?=$400 

<=9  years 

.-.  log.  s  =  2. 7553666 

log.jp=2.6020600 

.-.log.s— log.p=  .1533066 

.1533066 
log.(l+r)=  - 

=  .0170340 
=log.  of  1.04 
.•.  r=  .04= 4  per  cent. 

example  v. 

in  what  time  will  a  sum  of  money  double  itself,  allowing  4  per  cent,  com- 
i  mnd  interest  ? 

Here  s,  p,  r  are  given,  and  t  is  sought. 
From  the  formula  (7)  we  have 

s=p(l+r)\ 
But  here  s=2p 

.•.2j?=jp(l+r)1 
.-.2=(l+r)t 
log.  2 
'=Iog.(l  +  r) 
.3010300 


"".0170333 
=  17.673  years 
=  17  years,  8  months,  2  days. 
In  like  manner,  if  it  bo  required  to  find  in  what  tune  a  sum  will  triple  itself 
•t  the  same  rate,  we  have 

log.  3 
t  = 


log.  1.04 
.4771213 


.0170333 
=28.011  years 
=28  years,  o  months,  3  days. 


INTEREST  AND  ANNUITIES.  291 

PRESENT   VALUE   AND   DISCOUNT  AT    COMPOUND    INTEREST. 

If  we  call  p  tlio  present  value  of  a  sum  s  due  t  years  hence,  and  d  its  dis- 
•  count,  reasoning  precisely  in  the  same  manner  as  in  the  case  of  simple  inter- 
est, we  shall  find 

*=(IT^ (8) 

M1-^) <9> 

ANNUITIES   AT  COMPOUND   INTEREST. 

Problem  VIII. —  To  find  the  amount  of  an  annuity  a  continued  for  t  yeais, 
compound  interest  being  allowed  at  the  rate  r. 

At  the  end  of  the  first  year  the  annuity  a  will  become  due  ;  at  the  end  of 
the  second  year  a  second  payment  a  will  become  due,  together  with  the  in 
terest  of  the  first  payment  a  for  one  year,  that  is,  ar ;  the  whole  sum  upon 
which  interest  must  now  be  computed  is  thus,  2a-{-ar. 

At  the  end  of  the  third  year  a  further  payment  a  becomes  due,  together  with 
the  interest  on  2a-{-ar,  i.  e.,  2ar-\-ar^;  the  whole  sum  upon  which  interest 
must  now  be  computed  is  3a-\-3ar-\-ar2.     The  result  will  appear  evident 
when  exhibited  under  the  following  form  : 
Whole  amount  at  the  end  of  first  year,       =a. 
Whole  amount  at  the  end  of  second  year,  =za-\-a-\-ar 

=a  +  a(l+r). 
Whole  amount  at  the  end  of  third  year,    =a-\-a-\-a(l-{- r)-\-ar -\-ar{l-\-r) 

=  a  +  a(l  +  r)  +  a(l+ry: 
Whole  amount  at  the  end  of  fourth  year,  =a-\-a  -f-  a(l-\-r)  -j-  a(l-|-r)8  +  o.r 

+ar(l+r)  +  ar(l  +  r)2. 
=a+a(l+r)+a(l+r)-+a(l+r)3 
5cc.  &c.  &c. 

Whole  amount  at  the  end  of  th  year,  r=a4-o(l  +  r)+a(l  +  r)3+a(l-f -):i 

+ a(l+r)<-i. 

Hence  the  whole  amount  is,  in  terms  of  the  sum  of  a  geometric  progression 

5  =  ajl  +  (l  +  r)  +  (l+r)'+ +  (1  +  ^1 

S-^i — (10) 

Problem  IX. —  To  find  the  present  value  of  an  annuity  a  payable  for  X 
years,  compound  interest  being  allowed  at  the  rate  r. 

It  is  manifest  that  the  present  value  of  this  annuity  must  be  a  sum  such, 
that  if  put  out  at  interest  for  t  years  at  the  rate  r,  its  amount  at  the  end  of  that 
period  will  be  the  same  as  the  amount  of  the  annuity. 

Hence,  if  we  call  this  present  value  p,  we  shall  have,  by  Prcbs.  VII.  and 
VIII., 

p(l-^-r)l=  amount  of  annuity 

(i+rr-i 

=a. 

r 


1   r{l  +  ry 
a    (l-|_ r)«—  1 


P=   r(i  +  r).    -a 


(H) 


Now 


Also, 


292  ALGEBRA. 

EXAMPLE. 

What  is  the  present  value  of  an  annuity  of  $500,  to  last  for  40  years,  cow 
pound  interest  being  allowed  at  the  rate  of  2\  per  cent,  per  annum. 
By  formula  (11), 

_a    (1  +  r)1— 1 
i?  =  ;-     (1+r)1    " 

Here 

a =$500 
r  =$.025 
t  =40  years; 

.-.  (l+r)t=(1.025)*'. 

log.  (1.025)«=40  log.  1.025 
=  40  X  .0107239 
=  .4289560 
=  log.  2.685072 
.-.  (1.025)40=2.685072=(l+r)t. 

a      500 
-r=^5  =  2000° 

1.685072 
••^=20000X  2^85072 
=20000  X-62757... 
=  12551.40  dollars.    > 

REVERSION  OF  ANNUITIES. 

Problem  X. —  To  find  the  present  value  (P)  of  an  annuity  a  which  is  to  com 
mence  after  T  years,  and  to  continue  for  t  years. 

The  present  value  required  is  manifestly  the  present  value  of  a  for  T-f  f 

years,  minus  the  present  value  of  a  for  T  years. 

a    (l  +  r)T+l  — 1 
By  Problem  IX.,  the  present  value  of  a  for  T+*  years  =-  .  -  .  T+t 

„      ,       m  a     (!+r)T  — I 

By  Problem  IX.,  the  present  value  of  a  for  T  years        =- .     .         77 — 
P=".  \  (l+r)-T-(l+r)-<T+t>£     (12) 

PURCHASE  of  estates. 

Problem  XL — To  find  the  present  value  p  of  an  estate,  or  perpetuity,  whose 
annual  rental  is  a,  compound  interest  being  calctdated  at  the  rate  r. 

The  present  value  of  an  annuity  a,  to  continuo  for  t  years,  by  Prob.  IX.,  is 

p=\\l-{l  +  r)-<\; 

but  if  tho  annuity  last  forever,  as  in  tho  caso  of  an  estate,  then  t=  t\  and 

•.  - — ; — -  =  _=0  ;  hence,  in  tho  present  case, 

(I-)-?")'         OD 

P=°Z (W) 


INTEREST  AND  ANNUITIES.  293 

EXAMPLE. 

What  is  the  value  of  an  estate  whose  rental  is  $1000,  allowing  the  pur- 
chaser 5  per  cent,  for  his  money  ? 
Here 

a=$1000 
r  =  $.05 

1000 
•••  P=~^b 

==20000,  or  20  years'  purchase. 

REVERSION   OF   PERPETUITIES. 

Problem  XII. — To  find  the  present  value  of  an  estate,  or  perpetuity,  whost 
annual  rental  is  a  dollars,  to  a  person  to  whom  it  will  revert  after  T  years, 
compound  interest  being  allowed  at  the  rate  r. 

By  Problem  X.,  the  present  value  of  an  annuity,  to  commence  after  T  years, 
and  to  continue  for  t  years,  is 

p=£j(l+r)-*-(l+r)r(*wj 

In  the  present  case,  <=co  ,  and  .-.  (l+r)  ~(T+t>=0 ;  hence  we  shall  have 

P=7-(i+7jx (14> 

EXAMPLES  FOR  PRACTICE. 

(1)  Find  the  interest  of  3555  for  2^  years  at  4£  per  cent,  simple  interest. 

Ans.  $ 65.906. 

(2)  In  what  time  will  the  interest  of  $1  amount  to  75  cents,  allowing  4|  per 
cent,  simple  interest  ? 

Ans.  16  years,  8  months. 

(3)  What  is  the  amount  of  $120. 50  for  2|  years  at  4£  per  cent,  simple  in- 
terest ? 

Ans.  $134,809. 

(4)  The  interest  of  d£25  for  3|  years,  at  simple  interest,  was  found  to  be 
d€3  185.  9d. ;  required  the  rate  per  cent,  per  annum. 

Ans.  4|. 

(5)  Find  the  discount  on  d£100  due  at  the  end  of  3  months,  interest  being 
calculated  at  the  rate  of  5  per  cent,  per  annum. 

Ans.  c£l  4s.  8\d 

(6)  What  is  the  present  value  of  the  compound  interest. of  c£l00  to  be  re- 
ceived five  years  hence  at  5  per  cent,  per  annum  ? 

Ans.  <£78  7*.  0\d. 

(7)  What  is  the  amount  of  <£721  for  21  years  at  4  per  cent,  per  annum 

compound  interest? 

Ans.  .£1642  195.  $\d 

(8)  The  rate  of  interest  being  5  per  cent.,  in  what  number  of  years,  at  com- 
pound interest,  will  $1  amount  to  $100  ? 

Ans.  94  years,  141.4  days. 

(9)  Find  the  present  value  of  <^430,  due  nine  months  hence,  discount  being 

allowed  at  4i  per  cent,  per  annum. 

Ans.  d£415  19s.  2±d. 


294  ALGEBRA 

(10)  Find  the  amount  of  81000  for  1  year  at  5  per  cent,  per  annum,  coni- 
pound  interest,  the  interest  being  payable  daily. 

Ans.  $1051.288  nearly 

(11)  What  sum  ought  to  be  given  for  the  lease  of  an  estate  for  20  years,  of 

the  clear  annual  rental  of  £100,  in  order  that  the  purchaser  may  make  8  per 

cent,  of  his  money  ? 

Ans.  .£981  16s.  3^(1 

(12)  Find  the  present  value  of  £20,  to  be  paid  at  the  end  of  every  five  years, 

forever,  interest  being  calculated  at  5  per  cent. 

Ans.  £72  7s.  911. 

(13)  What  is  the  present  value  of  an  annuity  of  £20,  to  contiuue  forever, 
and  to  commence  after  two  years,  interest  being  calculated  at  5  per  cent.  ? 

Ans.  ,£362  16s.  2}d. 

(14)  The  present  value  of  a  freehold  estate  of  £100  per  annum,  subject  to 
the  payment  of  a  certain  sum  (A)  at  the  end  of  every  two  years,  is  £1000, 
allowing  5  per  cent,  compound  interest.     Find  the  sum  (A). 

Ans.  A  =£102  10s. 

(15)  What  is  the  present  value  of  an  annuity  of  £79  4s.,  to  commence  7 

years  hence  aud  continue  forever,  interest  being  calculated  at  the  rate  of  4), 

per  cent.  ? 

Ans.  £1293  5s.  U\d. 


INTERPOLATION. 

236.  This  name  is  applied  to  the  process  of  finding  intermediate  numbers 
between  those  given  in  tables. 

Tables  are  generally  calculated  from  an  algebraic  formula  in  which  there 
are  two  variable  quantities,  the  one  of  which  is  called  a.  function  of  the  other, 
the  latter  being  usually  called  the  argument  of  the  function. 

Thus,  logarithms  are  functions  of  the  numbers  to  which  they  belong,  the 
numbers  being  the  arguments.  Several  formulas  expressing  the  relation  be- 
tween a  number  and  its  logarithm  have  been  seen  by  the  student,  and  will 
serve  to  exemplify  the  formulas  in  general  of  which  we  are  now  speaking. 

The  substitution  of  successive  numbers  for  the  argument,  the  calculating  of 
the  corresponding  values  of  the  function,  and  writing  the  residts  in  a  table,  is 
called  tabulating  the  formula. 

If  the  formulas  which  have  been  derived  under  our  articles  upon  interest 
and  annuities  should  be  tabidated,  they  would  furnish  what  are  called  interest 
tables. 

The  function  frequently  depends  upon  two  arguments,  as  in  the  formula 
for  simple  interest, 

i=ptr (1) 

Here  the  function  is  i,  the  interest,  and  the  arguments  are, p  the  principal,  and 
r  the  rate.  This  requires  a  tablo  of  double  entry,  the  usual  form  of  which  is 
a  table  in  several  columns  occupying  the  whole  width  of  the  page,  the  ai  JU- 
ments  being  placed,  the  successive  values  of  the  one  in  a  horizontal  line  at  ihe 
heads  of  the  columns,  and  of  the  Other  in  a  vertical  line  at  the  side  of  the  | 
the  corresponding  values  of  the  function  being  placed  in  the  column  wader  ana 
uf  its  arguments,  and  on  the  horizontal  lino  of  the  other.     The  formula  (1) 


INTERPOLATION. 


295 


above  may  employ  a  table  of  triple  entry,  the  three  arguments  being  the  prin- 
cipal, the  rate,  and  the  time.  Such  a  table  is  formed  by  giving  a  whole  page 
to  the  argument  of  rate,  the  side  and  top  being  occupied  by  the  arguments  of 
principal  and  time. 

Where  the  differences  of  the  functions  are  proportional  to  the  differences 
of  their  arguments,  then  the  interpolation  is  made  by  simply  solving  a  pro- 
portion, the  first  two  terms  of  which  are  the  difference  of  the  tabulated  func- 
tions and  the  difference  of  their  arguments;  the  third  term  being  the  differ- 
ence between  one  of  the  tabulated  arguments  and  that  whose  function  is  to 
be  interpolated ;  the  fourth,  or  unknown,  term  of  this  proportion  will  be  the 
interpolated  function  required.  This  is  called  the  method  by  first  differences, 
and  has  been  exemplified  in  taking  out  logarithms  of  large  numbers  not  found 
exactly  in  the  tables. 

When  the  differences  of  the  functions  are  not  nearly  proportional  to  the 
differences  of  the  arguments,  as  in  the  case  of  the  logarithms  of  small  numbers, 
the  method  of  interpolation  above  described  would  not  be  sufficiently  accurate. 
The  nature  of  the  variation  of  the  function,  as  the  argument  varies  in  value,  is 
made  sensible  by  taking  the  difference  between  each  two  of  three  consecutive 
functions  in  the  table,  and  comparing  the  difference  between  the  first  and  sec- 
ond with  the  difference  between  the  second  and  third.  If  these  differences 
are  the  same,  we  have  seen,  in  the  note  to  (Art.  233),  that  the  method  of  first 
differences  already  explained  applies  ;  but  if  they  are  not,  their  difference, 
which  is  called  a  second  difference,  will,  by  its  magnitude,  indicate  the  degree 
of  inaccuracy  of  the  method  of  first  differences.  This  exposition  will  serve  to 
exhibit,  in  a  general  way,  the  nature  and  office  of  second  differences.  We 
proceed  to  give  a  more  analytic  development  of  the  use  of  second,  third,  &c. 
differences,  the  latter  holding  ths  same  relation  to  the  second  differences  that 
these  do  to  the  first. 

Let/and/'4"('i  represent  two  consecutive  functions  in  the  table,  Jj  being 
their  first  difference.  The  next  consecutive  function,  if  the  first  differences 
were  constant,  would  be  expressed  byf-{-2Sl;  but  as  they  are  supposed  not 
to  be,  it  must  be  expressed  by  the  formf-\-2Sl-\-d2,  c52  being  the  second  dif- 
ference, or  difference  between  the  two  first  differences,  (5,  and  ^,-j-^,.  The 
scheme  below  will  show  the  form  of  the  successive  functions  : 

Function* 
/ 

/+4<J,  +  6<J2+4<J3-H4 

and  so  on ;  from  which  we  perceive  that  the  coefficients  are  the  same  as  m  the 
expansion  of  a  binomial,  that  of  the  second  term  being  the  number  of  the  con- 
secutive functym  after  the  first  function.  Denoting  this  number  by  n,  we 
have  for  the  general  form  of  the  nth  function  after  the  first, 

-l)(n-2), 


1st  Differences. 

2d  Differences. 

3d  Differ- 
ences. 

4th  Dif 
Terence* 

t5,-|-f52 

<52 

<52  +  <53 
da+2<J3+34 

<>3  +  <'4 

^ 

n(n  — ')        n(n- 


-*3+  •••  +  *. 


[C] 


1.2    "a  '  1.2.3 

Suppose,  now,  that,  a  value  of  the  function  intermediate  between  the  first  and 

second  of  the  series  in  the  table  bo  required,  n  here,  instead  of  be/ng  an  entire 

number,  is  a  fraction.     If  the  value  of  the  function  be  required,  corresponding 


m 


ALGEBRA. 


to  a  value  of  the  argument  midway  between  its  consecutive  values  in  the  tabic 
n  becomes  equal  to  -.     If  the  arguments  of  the  tables  differ  by  24  hours,  and 

3        1 

the  function  be  required  for  3  hours,  n  becomes  equal  to  — ,  or  -.     If  the  tabu 

lar  arguments  differ  by  1  hour,  or  60  minutes,  and  the  function  be  required  for 

15      1 
an  argument  15  minutes  beyond  an  even  hour,  n=—=-. 


EXAMPLE. 

Given  the  logs,  of  15,  16,  17,  18,  19,  to  find  that  of  17.25. 


Arg  or  No. 

Func.  or  Log. 

1st  Difs.  5 1. 

2d  Difs.  S». 

3d  Dife,  r':i. 

,54. 

15 
16 

17 
18 
lit 

1.17609126 
1.20411998 
1.23044892 
1.25527251 
1.27875360 

2802872 
2632894 
2482359 
2348109 

—  169978 

—  150535 

—  134250 

+  19443 
+  16285 

—  3158 

The  numbers  in  the  third  column  are  obtained  by  taking  the  differences  of 

the  consecutive  numbers  in  the  second.     The  numbers  in  the  fourth  column 

from  the  second  in  the  same  way. 

9  9 

As  2.25  is  -  the  interval  between  15  and  18,  we  make  n=-,  and  have  for 
4  4 

formula  (C),  taking  <J1=2802872,  <l=  —  69978,  (53=19443,  <54=—  3158. 

The  result  would  be  nearly  the  same  by  neglecting  <54  and  using  the  mean  of  the  two 
third  differences* 


f= 

1.176126 

n6l  =  9-6- 

306462 

"(n  —  1)*  _45,<  _ 
1.2      2      32  s 

—  239031 

n(n  —  l)(n—2)A  __  45  A  _ 
1.2.3          3  — 384  3 

2278 

n(n- 

-l)(n-2)(n-3)A            135  A  _ 
1.2.3.4              *          6144"  * 

69 

Value  of  func.  required,  viz.,  log.  17.25  =  1.23678904 
The  formula  for  interpolation  may  be  derived  very  elegantly  by  the  method 
of  indeterminate  coefficients.  Thus,  let  y  represent  the  value  of  the  interpo- 
lated function  to  be  found,  A  the  argument  in  the  table,  m  the  number  of  parts 
(4tl'8  in  the  example  above)  between  A  and  the  consecutive  argument  of  the 
table,  and  n  the  whole  number  of  parts  (4  in  the  above  example)  between 
these  consecutive  arguments.  It  is  evident  that  y,  depending  on  A  and  m. 
may  be  expressed  in  terms  of  these.     Assume,  therefore, 

?/  =  A4-Bm4-Cm!4-Dm+,  &c., 
in  which  B,  C,  D,  &c,  are  undetermined  coefficients,  whose  values  are  to  be 
found. 

Now  let  m  have  successive  values,  represented  by  0,  n,  2n,  3w,  &c,  then 
the  corresponding  values  of  y  will  be 

*  As  means  are  much  used  in  calculations  with  tables,  it  may  be  well  to  advertise.  th« 
student  that  a  mean  of  three  numbers  is  obtained  by  adding  diem  together  and  dividing  by 
3  ;  of  five  numbers,  by  adding  thorn  together  and  dividing  the  sum  by  5,  an*  «o  on. 


INTERPOLATION.  297 

A (1) 

A  +  Bn4-Cn2+Dn3+,&c (2) 

A+B.2n+C(2ra)2+D(2rc)3+,&c (3) 

A+B.3tt  +  C(3w)2+D(3n)3+,&c (4) 

&c. 
Subtracting  successively  (1)  from  (2),  (2)  from  (3),  &c,  and  representing 
the  remainders  by  P',  Q',  R/,  &c,  and  dividing  by  n,  we  have 

P' 

—  r=B  +  C.n+D>i2+,&c (5) 

Q' 

— =B-f  C.3/i  +  D7»2+,&c (6) 

IV 

R' 

— =B  +  C.5»+Dl9?i2+,&c (7) 

&c.  &c. 

Again,  subtracting  successively  (5)  from  (6),  (C)  from  (7),  &c,  and  repre- 
senting the  remainders  by  P",  Q",  &c,  and  dividing  by  2n,  we  get 

^=C+D.3»+,&c (8) 

^=C  +  D.G«+,&C (9) 

&c.  &c. 

Next,  subtracting  (8)  from  (9),  &c,  and  representing  the  remainder*  Uy  P"\ 
&c,  and  dividing  by  3n,  we  have 

^=D  +  '&C <10> 

0"— P"          _      R'— Q'         „      Q'— P' 
But  T'"=^- ;  alsoQ"  = —  and  P"=- ; 

2n  n  n 


i 


(R'-Q')-(Q'-P') 


"~  2ra2 


Putting  d3  for  the  numerator  of  this  fraction,  we  have  by  (10), 

n— — —- ^2. 

3/i       6/i3' 
Substituting  this  value  of  D  in  (8),  and  transposing,  there  results 

r— —    ^l 

2w—2ra2' 

Q'— P' 
But  P"=— ,  and  putting  d2  for  Q'— P',  we  obtain 

2n2     2ra2* 
Again,  substituting  these  values  of  D  and  C  in  (5),  and  transposing,  we  have 

„       P'       do        d3        <*3 

n      2nr2n     6«.' 

or,  putting  d,  for  P',  and  simplifying, 

d,       do       do 

n      2n     3n 

Finally,  substituting  these  values  of  the  coefficients  B,  C,  D  ...  in  the  as 

sumed  equation,  we  obtain 

to         1  m  (m       \  1  ?«/m2     3m  „     \  , 

w=A+-d,  +  -  -(-_i)(5„+--(— 4-2)d3  +  ,  &c, 

*  '  n       '  2  n\n        I   -  '  6  n  \  n2       n    '     /  a  ' 


298 


ALGEBRA. 


as  the  formula  for  interpolation,  which  coincides  with  the  one  obtained  before 
^i»  ''si  ^3  ••  being  the  first,  second,  and  third  differences  of  the  functions,  a,  is 
evident  from  the  manner  in  which  tiny  have  been  assumed  above. 

Let  us  apply  it  to  a  table  in  the  Nautical  Almanac,  which  gives  the  moon's 
latitude  at  noon  and  midnight  for  every  day  in  the  year. 

EXAMPLE. 

Let  it  be  required  to  find  the  moon's  latitude  for  August  4,  1842,  at  16"  18" 
mean  time  at  Greenwich,  that  is,  at  4.3  hours  after  midnight. 


Moon's  Latitude. 

Si- 

Oo* 

Mean  Secocd  Difference. 

o      '          " 

Aug.  4.  Noon,  +0  45  48.1 
Midni-rht,  +0    5  54.6 

Aug.  5.  Noon,*  —0  34  33.1 
Midnight,  —1  14  49.4 

—  39  53.5 
—40  27.7 
—40  16.3 

+  34.2 
—  11.4 

+  11.4 

Now,  to  apply  the  formula,  we  have 

A=0°  5'  54".6,  (5,  =  — 40'  27".7,  or  —40.463  minutes; 

m     4.3  m 

-=—=0.358,  -<5.  =  — 14'  29".16; 
12  n    l 


n 


m                                mlm       \  . 
o\  =  +  ll".4,  — 1  =  —  0.642,  i  -(- l)<J.r=— 1".31. 

1  n  -  n  \n        I    - 

Therefore,  y  = — 0°  8'  35".87,  which,  without  the  sign  — ,  is  the  moon's 
correct  latitude  south  at  the  time  for  which  it  was  required. 

Second  differences  will  ordinarily  insure  sufficient  accuracy.  Third  and 
fourth  differences  are  rarely  used. 


INEQUATIONS. 

237.  In  discussing  algebraical  problems,  it  is  frequently  necessary  to  intro- 
duce inequations,  that  is,  expressions  connected  by  the  sign  >.  Generally 
speaking,  the  principles  already  detailed  for  the  transformation  of  equations 
are  applicable  to  inequations  also.  There  are,  however,  some  important  ex- 
ceptions which  it  is  necessary  to  notice,  in  order  thai  the  student  may  guard 
against  falling  into  error  in  employing  the  sign  of  inequality.  These  excep- 
tions will  be  readily  understood  by  considering  the  different  transformations  in 
succession. 

I.  If  we  add  the  same  quantity  to,  or  subtract  it  from,  the  two  members  of  any 
inequation,  the  resulting  inequation  will  always  hold  good,  in  the  same  sense 
as  the  original  inequation  ;  that  is,  if 

a~>b,  then  a+a'>t  +  a',  and  a  —  n'>b— a'. 

Thus,  if 

8>3,        wo  have  still       8+5>.'+'>,       and       8— 5>3— 5. 
So,  also,  if 

— 3<— 2,  wo  havo  still  —  3  +  6 <—  2+6.  and  —  3— 6<—  2  —  fc.f 

,*  Tin'  i> n'l  latitude  is  marked  +  when  north,  —  when  south. 

t  Tin-  negative  qt  iter  numerical  « al  te  ia  ata  :..  -  lered  less  titan  the 

negative  quantity  ol' 1'>S  numerical  \ 


INEQUATIONS.  209 

The  truth  of  tliis  proposition  is  evident  from  what  has  been  said  with  refer- 
ence to  equations. 

This  principle  enables  us,  as  in  equations,  to  transpose  any  term  from  one 
member  of  an  inequation  to  the  other  by  changing  its  sign. 
Thus,  from  the  inequation 

a°~+  b°~>3b-—2a2, 
we  deduce 

a2+2a2>3&2—   62, 
or 

3a2>262. 

II.  If  we  add  together  the  corresponding  members  of  two  or  more  inequations 
which  hold  good  in  the  same  sense,  the  resulting  inequation  will  always  hold 
good  in  the  same  sense  as  the  original  individual  inequations  ;  that  is,  if 

a>&,  c>d,  e>/, 
then 

a  +  c+e>b+d+f. 

III.  But  if  we  subtract  the  corresponding  members  of two  or  more  inequations 
which  hold  good  in  the  same  sense,  the  resulting  inequation  will  not  always 
hold  good  in  the  same  sense  as  the  original  inequations. 

Take  the  inequations  4<7,  2<3,  we  have  still  4—  2<7  —  3,  or  2<4. 

But  take  9<10  and  6<8,  the  result  is  9— 6>  (not  <)  10—8,  or  3>2. 

We  must,  therefore,  avoid  as  much  as  possible  making  use  of  a  transforma- 
tion of  this  nature,  unless  we  can  assure  ourselves  of  the  sense  in  which  the 
resulting  inequality  will  subsist. 

IV.  If  we  multiply  or  divide  the  two  members  of  an  inequation  by  a  positive, 
quantity,  the  resulting  inequation  will  hold  good  in  the  same  sense  as  the  original 
ineauation.     Thus,  if 

a      b 
a<,b,      then      ?na<^mb,         —  <- 


-a^> — b,  then  — na^> — nb, 


m     m 
a         o 
n^     n 

This  principle  will  enable  us  to  clear  an  inequation  of  fractions. 

Thus,  if  we  have 

a-—b"-     c^—d- 


> 


2d    <■    3a    ' 

multiplying  both  members  by  Gad,  it  becomes 

3a{a?—¥)>2d(c'2—di). 
But, 

V.  If  we  multiply  or  divide  the  two  members  of  an  inequation  by  a  negative 
quantity,  the  resulting  inequation  will  hold  in  a  sense  opposite  to  that  of  the 
original  inequation. 

Thus,  if  we  take  the  inequation  8>7,  multiplying  both  members  by  — 3, 
we  have  the  opposite  inequation,  — 24  < — 21. 

8  7  8  7 

Similarly,  8>7,  but  ^^<-^$  or  —3  <  —  3" 

VI.  We  can  not  change  the  signs  of  both  members  of  an  inequation  unless  we 
reverse  the  sense  of  the  inequation,  for  this  transformation  is  manifestly  the  same 
thing  as  multiplying  both  members  by  — 1. 


300  ALGEBRA. 

VII.  If  both  members  of  an  inequation  be  positive  numbers,  we  can  raise  them 
to  any  power  without  altering  lite  sense  of  the  inequation  ;  that  is,  if 

a>l,  then  an>&\ 
Thus,  from  5>3  wo  have  (5)2>(3)2,  or  25>9. 

So,  also,  from        (a+ty^c,  we  have  (a-|~^):>cs. 
But, 

VIII.  If  both  members  of  an  inequation  be  not  positive  numbers,  we  can  not 
determine,  a  priori,  the  sense  in  which  the  resulting  inequation  will  hold  good, 
unless  the  power  to  which  they  are  raised  be  of  an  uneven  degree. 

Thus,  — 2<3       gives  (— 2)2<      (3):,  or         4<9; 

But,  _3>_5  gives  (—3)2<(  —  5)2,  or        9<25; 

Again,         — 3>—  5  gives  (— 3)3>(  — 5)3,  or  —  27>— 125, 
In  like  manner, 

I  X.    We  can  extract  any  root  of  both  members  of  an  inequation  wiOioul  alter 
ing  the  sense  of  the  inequation  ;  that  is,  if 

a>6,  then  V*a>  Vb. 
If  the  root  be  of  an  even  degree,  both  members  of  the  inecuation  must 
necessarily  be  positive,  otherwise  wo  should  be  obliged  to  introdu  :e  imaginary 
quantities,  which  can  not  be  compared  with  each  other. 

EXAMPLES   IN   INEQUATIONS. 

(1)  The  double  of  a  number,  diminished  by  G,  is  greater  than  24  ;  and  triple 
the  number,  diminished  by  G,  is  less  than  double  the  number  increased  by  10. 
Required  a  number  which  will  fulfill  the  conditions. 

Let  x  represent  a  number  fulfilling  the  conditions  of  the  question  ;  then,  in 
the  language  of  inequations,  we  havo 

2x— 6>24,  and  3.r— G<2x+10.  • 
From  the  former  of  these  inequations  we  have 

2x>30,  or  .r>15; 
and  from  the  latter  we  get 

3x— 2x<10  +  G,  or  a:<lG; 
therefore  15  and  16  are  the  limits,  and  any  number  betwcoa  these  limits  will 
satisfy  the  conditions  of  the  question.     Thus,  if  we  take  the  number  15-9,  we 
nave 

15-9X2  — G>21  by  1-8, 
while  15-9X3  — 6<15-9X2+10  by  0-1. 

5       4 
(2)  3x-2>^-- 

.-.  30r— 20>25.r— 8 
30z— 25x>20  —  8 
6x>12 
12 
X>~5- 


3)  43— 5.r<10— 8x. 


7     5 
(4)  6-4*<8-2.r. 


Ans.  r< — 11. 


A 

An-,  r-     -. 


INEQUATIONS.  30J 

12 
In  tlie  second  example,  — ,  or  2|,  is  an  inferior  limit  of  the  values  of  r. 

82 
In   the  tthird,  — 11,  and,  in  the  fourth,  — ,  or  9^,  are  superior  limits  of  the 

value  of  x.  If  the  second  and  fourth  of  the  above  inequalities  must  be  verified 
simultaneously  by  the  values  of  x,  these  values  must  be  comprised  between 
2§  and  9£.  If  the  third  and  fourth,  it  is  sufficient  that  it  be  less  than  — 11. 
Finally,  there  is  no  value  which  will  verify  at  the  same  time  the  2°  and  3°. 

(5)  3x— 2;y>5,  5z+3i/>lG; 

<5+2y           ^  16-3y 
.-.  ar> — —  and  a:> — . 

We  can  attribute  to  y  any  value  whatever,  and  for  each  arbitrary  value  ol 
y  we  can  give  to  x  all  the  values  greater  than  the  greatest  of  the  two  quan- 
tities 

5+2?/   16—3i/ 
— 3~~ '  ~~ 5~ " 
We  determine,  also,  from  the  proposed  inequalities, 

^3x— 5      ^  16— bx 

K-g-'  y>— — • 

In  order  that  these  last  two  may  be  fulfilled, 

Zx—h     16— 6a; 

47 
Thus  x  can  receive  only  values  superior  to  — ,  or  2T9g,  and  for  each  value 

of  x  there  should  be  admitted  for  y  but  values  comprised  between  the  two 
limits  above. 

(6)  ar34-4x>12 
.-.  x2+4ar+4>16 

z+2>±4 

r>2,  or  —2. 
The  inferior  limit  of  x  is  -}-2. 

(7)  z2+7a;<30. 

Ans.  x<C3  or  — 10. 

The  superior  limit  of  a:  is  — 10. 

P 

(8)  Reduce  £n>£n to  its  most  simple  form. 

Ans.  x^>p-\-1 

j.n-r+1 

(9)  Reduce  a:n>_p —  to  its  most  simple  form. 

Ans.  r>l-f-  \Jp. 


302  ALGEBRA. 


GENERAL  THEORY  OF  EQUATIONS. 

THE  NATURE  AND  COMPOSITION  OF  EQUATIONS 

238.  The  valuable  improvements  recently  made  in  the  process  for  the  de- 
termination of  the  roots  of  equations  of  all  degrees,  render  it  indispensably 
necessary  to  present  to  the  student  a  view  of  the  present  statg  of  this  interest- 
ing department  of  analytical  investigation.  The  beautiful  theorem  of  M.  Sturm 
for  the  complete  separation  of  the  real  and  imaginary  roots,  and  for  discover- 
ing their  initial  figures,  combined  with  the  admirable  method  of  continuous 
approximation  as  improved  by  Horner,  has  given  a  fresh  impulse  to  this  branch 
of  scientific  research,  entirely  changed  the  state  of  the  subject,  and  completed 
the  theory  and  numerical  solution  of  equations  of  all  degrees. 

We  recapitulate  here  two  or  three 

DEFINITIONS. 

1.  An  equation  is  an  algebraical  expression  of  equality  between  two  quan- 
tities. 

2.  A  root  of  an  equation  is  that  number,  or  quantity,  which,  when  substi- 
tuted for  the  unknown  quantity  in  the  equation,  verifies  that  equation. 

3.  A  function  of  a  quantity  is  any  expression  involving  that  quantity  ;  thus. 

ax1  +  b 
ax--\-b,  flr'-f-cx-f-rf,  ,,  a1  are  all  functions  of  x ;  and  also  ax2  — by9, 

. 2x-f3y 

V4x— 5?/,   3x— 2f  2/2+2/*+a;S!+a2+&+2,  are  all  functions  of  x  and  y. 

These  functions  are  usually  written/(x),  and/(x,  y). 

4.  To  express  that  two  members  of  an  equation  are  identical  or  true  for 
every  value  of  x,  the  sign  az  is  sometimes  used. 

PROPOSITION   I. 

Any  function  ofx,  of  the  form 

xn-\-j)xa~l  -f  qxn^-\-  rx"-3-f- 

when  divided  by  x— a,  will  leave  a  remainder,  which  is  the  same  function  of  a 
that  the  given  polynomial  is  ofx. 

Let  f(x)=xn+pxa-l  +  qx"-"-+ ;  and,  dividing/(x)  by  x— a,  let  Q  de- 
note the  quotient  thus  obtained,  and  R  the  remainder  which  does  Dot  involve 
x  ;  hence,  by  the  nature  of  division,  we  have 

/(*):rQ(z-a)+R. 

Now  this  equation  must  be  true  for  every  value  ofx,  because  its  truth  de- 
ponds  upon  a  principle  of  division  which  is  independent  of  the  particular  values 
of  the  letters;  hence,  if  x=a,  we  have 

*/(a)  =  0  +  R; 
and,  therefore,  the  remainder  R  is  the  same  function  of  a  that  the  proposed 
polynomial  is  of  x. 

EXAMPLES. 

(l)  What  is  the  remainder  of  x-  —  Gx+7.  divided  byx—2,  without  actually 
performing  the  operation  ? 


*  The  rtndenl  will  recollect  tliatyfa)  stands  for  xn+px—l+,  $p.,  ami  feat,  therefore, 

I  lor^-f/,,"    X-\-qnn    "-f.  fee. 


GENERAL  THEORY  OF  EQUATIONS.  J3 

(2)  What  is  the  remainder  of  x3— 6x"-\-8x— 19,  divided  by  r+3  1 

(3)  What  is  the  remainder  of  x4-\-6x*-{-7x'2-\-5x— 4,  divideo  by  a:— 5  1 

(4)  What  is  the  remainder  of  z*-\-px2-\-qx-\-r,  divided  by  £ — al 

ANSWERS. 

(1)  R=22  —  6x2+7  =  —  1. 

(2)  R=(—  3)3— 6(  —  3)2+8(— 3)  — 19  =— 124. 

(3)  1571. 

(4)  a?-\-pa'i-{-qa-\-r. 

PROPOSITION  II. 

if  a  is  the  root  of  the  equation, 

xa+ A^"-^  A2xn-2+ An-^-r-  A„_1j4-An=0, 

the  first  member  of  the  equation  is  divisible  by  x — a. 

If  the  division  be  perfoi-rned,  the  remainder,  according  to  the  preceding 
proposition,  must  be  of  the  form 

an+Aian_1-|- A2an_2 {•  An_2a2+  An_ia-j-An  ; 

i.  c.,  the  same  function  of  a  that  tho  first  member  of  the  proposed  equation  is 
of  x ;  and,  therefore,  since  a  is  a  root  of  the  equation,  the  remainder  vanishes, 
and  the  polynomial,  or  first  member  of  the  equation,  is  divisible  exactly  by 
x — a. 

Conversely,  if  the  first  member  of  an  equation  f(x)=0  be  divisible  by  x  —  a, 
then  a  is  a  root  of  the  equation. 

For,  by  the  foregoing  demonstration,  the  final  remainder  is  f(a)  ;  but  since 
f(x),  or  the  first  member  of  the  equation,  is  divisible  by  x — a,  the  remainder 
must  vanish;  hence /(a) =0  ;  and  therefore,  a  being  substituted  for  x  in  the 
equation  /(.r)  =  0,  verifies  the  equation,  and,  consequently,  a  is  a  root  of  the 
equation. 

PROPOSITION    III. 

239.  The  proposition  that  every  equation  has  a  root,  has  in  most  treatises 
on  Algebra  been  taken  for  granted.  It  has,  however,  of  late  years  been 
thought  to  require  a  demonstration,  and  we  add  one  which  is  as  brief  and  clear 
as  any  of  the  best  modifications  of  that  by  Cauchy. 

As  it  will  prove  a  little  tedious,  the  student  may,  if  he  please  to  admit  the 
proposition,  pass  on  to  Prop.  IV. 

It  will  be  necessary  to  premise  a  few  lemmas  relating  to  the  properties  of 
moduli,  some  of  which  have  been  already  demonstrated  (Art.  197),  but  we  re- 
peat them  here  for  convenience  of  reference. 

Lemma  I. —  The  sum  or  difference  of  any  two  quantities  whatever  has  a 
modulus  comprehended  between  the  sum  and  difference  of  the  moduli  of  the 
two  quantities. 

Lemma  II. — The  modulus  of  a  product  of  two  factors  is  equal  to  the  product 
of  their  moduli. 

Corollary. — Hence  the  product  of  the  moduli  of  any  number  of  factors  is 
the  modulus  of  their  product,  and  the  modulus  of  the  ft,h  power  of  a  quantity 
is  the  n"1  power  of  its  modulus. 

Lemma  III. — In  order  that  a  quantity  of  the  form  a+b\/ — 1  may  be  zero, 
it  is  necessary,  and  it  is  sufficient,  that  its  modulus  should  be  zero  ;  for  a  and 
b  being  real  quantities,  let 


304  ALGEMIA. 


a  +  by/  — 1=0. 

As  the  real  part  a  can  not  destroy  the  imaginary  part  b  y/  —  1,  we  must 
have  separately  <z  =  0  and  6  =  0  .-.  \/a--\-b-=Q. 

Lemma  IV — Let  there  be  a  polynomial  of  the  form 

X=xm— 2->xm~l  —  qxm~2 u, 

in  which  the  coefficients  of  all  the  terms  after  the  first  are  essentially  nega- 
tive. A  value  of  x  can  always  be  found  sufficiently  great  to  render  the  first 
term  xm  greater  than  all  the  others  together,  and,  consequently,  the  expression 
X  essentially  positive,  and  as  great  as  we  please. 

For  we  can  write  X  thus,  / 


\        x      xs  xmJ 


in  which,  if  x  be  supposed  to  increase  indefinitely,  the  negative  terms  in 
the  parenthesis  will  decrease  indefinitely.  As  soon  as  x  has  attained  a  value 
"K  sufficiently  great  to  make  these  negative  terms  together  equal  to  1,  the 
value  of  the  expression  X  will  go  on  increasing  indefinitely,  and  be  always 
positive. 

If  A  be  taken  negatively  instead  of  positively,  X  will  still  be  positive,  provided 
m  be  even ;  but  if  m  be  odd,  then,  when  —A  is  put  for  x,  the  leading  term  will 
be  negative,  and,  consequently,  X  negative. 

Corollary. — If  the  first  tevmp  of  a  series  p-\- qx-\- rx-+,  &c,  be  constant, 
x  may  be  taken  a  sufficiently  small  fraction  to  make  the  sign  of  the  whole  de 
pend  on  that  of  the  first  term.* 


*  From  the  above  it  may  be  shown,  that  in  every  equation  of  an  odd  degree  two  values 
can  always  be  found,  which,  when  separately  substituted  for  the  unknown  quantity,  will 
furnish  two  results  with  opposite  signs,  and  that  in  every  equation  of  an  even  degree 
two  such  values  can  also  be  assigned,  whenever  the  final  term  or  absolute  number  is 
negative;  for,  in  this  case,  the  substitution  of  zero  for  x  will  give  a  negative  result,  viz., 
the  absolute  number  itself,  and  the  substitution  of  -f-A  or  —1  will  give  a  positive  result. 

From  these  inferences  it  may  be  proved,  without  difficulty,  that  every  equation  of  an 
odd  degree,  without  exception,  has  a  real  root,  and  every  equation  of  an  even  degree,  pro- 
vided its  final  term  be  negative,  has  two  real  roots,  the  one  positive,  the  other  negative. 
This  conclusion  might  be  deduced  immediately  from  what  has  just  been  established,  if  it 
be  conceded  that  every  polynomial/(x),  which  gives  results  of  opposite  signs  when  twe 
values  a,  b  are  successively  given  to  x,  passes  from/(c)  to/(i)  continuously  through  all  in- 
termediate values,  as  x  passes  continuously  from  a  to  b.  But  this  is  a  principle  that  re- 
quires demonstration.    We  proceed  to  establish  it  with  the  necessary  rigor. 

PROPOSITION. 

If  in  the  polynomial 

/(x)=:rn+An_ia:n-1 ....  +Ajfi+Alas+TX 

x  be  supposed  to  vary  continuously  from  z=a  to  x=b,  then  the  function  f(x)  will  varv 
continuously  from /(a)  to  f(b). 

DEMONSTRATION. 

Let  </'  be  any  value  intermediate  between  a  and  b.    Substitute  af-\-h  for  x  in  the  pofo 

nomial.  ami  it  will  become 

f(a'+h)={a'+hy+Aa_l(a'+h)*-1 ....  A,K+ A)-+A,  (a'+AJ+N; 

that  is,  actually  developing,  in  the  second  member,  by  the  binomial  theorem,  and  ami  . 
the  r.sulis  according  to  tho  ascending  powers  of  A, 


GENERAL  THEORY  OF  EQUATION'S 


305 


1'RELIMINARY   DEMONSTRATION. 

240.  Each  of  the  equations 

has  a  root  of  the  form  a-\-l>  V — 1-  This  is  true  of  the  equation  .r,r'  — -j-li 
whether  m  be  even  or  odd,  sinco  x=l  always  satisfies  it.  It  is  also  true  of 
the  equation  xm= — 1  when  m  is  odd,  for  then  .r= — 1  satisfies  it. 

When  m  is  even,  it  must  either  bo  some  power  of  2,  or  else  some  power 
of  2  multiplied  by  an  odd  number ;  if  it  be  a  power  of  2,  then  the  value  of  x 
will  be  obtained  after  the  extraction  of  the  square  root  repeated  as  many  times 
in  succession  as  there  ore  units  in  the  said  power.  Now  the  square  root  of 
the  form  a-\-b  •/  —  1  is  always  of  the  same  form  (Art.  118).  Hence,  when 
m  is  a  power  of  2,  each  of  the  equations  xm=  —  1,  xm=±  -\f — 1  has  a  root 
of  the  form  announced.  When  m  is  a  power  of  2  multiplied  by  an  odd  num- 
ber, then,  if  we  extract  the  root  of  this  odd  degree  first,  there  will  remain  to 
be  extracted  only  a  succession  of  square  roots. 

We  have,  therefore,  merely  to  show  that,  when  m  is  odd,  a  root  of  rl-  V  —  1 
is  of  the  predicted  form. 

Now  the  o<M  powers,  1,  3,  5,  &c,  of  -f  V — 1>  are  (Art.  66) 


+  V—  if.—  V—  i.  +  V— !••• 


and  the  same  powers  of  —  V  —  1  are 


—  V  — 1.  +V—  h  —  V—  1 


<»nsequently,   when   m    is    odd,    a   root    of   i  V — 1  is  either  -{-  y/ — 1  or 
-  -/  —  1.     Hence  the  predicted  form  occurs,  whether  m  be  odd  or  even. 
It  follows  from  this  proposition  that,  whatever  positive  whole  number  m 


maybe,  ( —  l)m  and  ( yf — l)m   will  always  bo  of  the  form  a-\-b  sj — 1;  or, 


more  generally,  ( —  1)'"  and  ( -/  —  l),n  will  always  be  of  this  form,  n  and  m  be 
ing  any  integers  positive  or  negative  (Cor.  to  Lemma  II.). 

THEOREM. 

241.  Every  algebraical  equation,  of  whatever  degree,  has  a  root  of  the  form 


+An_1a'n-,-f(«-l)A„_1«'"^ 


-\-n{n — \)af 


+(»-l)(»-2)An_1«' 


/ 

/n— 3 


-{-Ata'i 
+A,a' 
+N 
which  mny  be  written 


?-+fc° 


..hn. 


Now,  by  what  has  been  above  shown,  a  value  so  small  may  bt  given  to  h  that  the  sum 
of  the  terms  al"ter/(a')  shall  be  less  than  any  assignable  quantity,  however  small.  Hence, 
whatever  intermediate  value  a'  between  a  and  b  be  fixed  upon  for  .r  iny(.r),  in  proceeding 
to  a  neighboring  value,  by  the  addition  to  a'  of  a  quantity  h  ever  so  minute,  we  obtain  fot 
f{af-\-h)  a  like  minute  increase  of  the  preceding  value  f \af).  In  other  words,  in  proceed- 
ing continuously  from  a  to  b  in  our  substitutions  for  x,  the  results  of  those  substitutions 
must  be,  in  like  r  anner.  continuous,  or  all  connected  together  without  any  unoccupied  in- 
terval. 

u 


306  A.LGEBRA. 

a-\-b^/  —  1.  whether  the  coefficients  of  the  equation  be  =  1 1 1  real,  oi  any  of 
them  imaginary  and  of  the  same  form. 

Let/(r)  — r"-|-  \       ,•   '+...A:,r1+A,j--fA,/-(-.\  =  il (1) 

represent  nny  equation  the  coefficients  of  which  are  either  real  or  imagiuary. 

It"  in  this  equation  we  substitute  /,-^-,/y/ —  i  for  r,  p  and  q  being  real,  the 
first  member  will  furnish  a  result  of  the  form  l'  +  <i  V  —  '•  P  ;"1''  vi  being 
real  (Lemma  II.).  Should  i>-\-qy/ —  1  be  ;i  root  of  tin1  equation,  this  r< 
must  be  zero  ;  or,  which  is  the  same  thing,  the  modulus  of  1'  +  ^  \/  — L  v'z' 
V^'+Q1'  must  l>e  zero  (Lemma  111.).  Ami  we  have  now  to  prove  that 
values  of/'  and  q  always  exist  that  will  fulfil]  this  latter  condition. 

In   order  to  this,   it   will  be  sufficient   to  show  that   whatever  value  of 
y/  |»  -_|_(  £-,  greater  than   zero,  arises   from   any   proposed    values   ofjJ   aid  q, 
other  values  of  ]>  and  7  necessarily  exist,  for  which    -v/P'+'i  still 

smaller,  so  that  the  smallest  value  of  which  ^ps-j-Q9  is  capable  must  be  zero, 
and  the  particular  expression p-f-0  V —  1,  whence  this  value  has  arisen,  must 
be  a  root  of  the  equation. 

For  the  purpose  of  examining  the  effect  upon  any  function, /(x),  of  eha 
introduced  into  the  value  of  x,  the  development  exhibited  at  Art.  239,  Note,  is 
very  convenient.     By  changing  x  into  .r-\-h,  the  altered  value  of  the  function  is 
thus  expressed  by 

/(r+70=/(.r)+/1(x)/i+/,(x)^+/;!(.r)1-^...^ (2) 

where /(.r)  is  the  original  polynomial,  and  J\(x),  /-(')•  &C.,  contain  none  but 
integral  and  positive  powers  of  x  (Art.  239,  Note). 

The  first  of  these  functions, /(x),  becomes  P-f-Q  yj  —  1  w\\ex\  p-\-q^ —  1 
is  substituted  for  x;  the  other  functions  may  some  of  them  vanish  for  the 
same  substitution,  for  aught  we  know  to  the  contrary  :  but  till  the  terms 
f(.v)  can  not  vanish  ;  the  last  Jin,  which  does  not  contain  x,  must  necessarily 
remain. 

Without  assuming  any  hypothesis  as  to  what  terms  of /(r-|-/()  vanish  for 
the  value  x=p-\-q^ —  1,  which  causes  the  first  of  those  terms, /(.r),  to  be 
come  P  +  Q  V  — li  lot  °8  represent  by  hm  the  least  power  of  h  for  which  the 
coefficient  does  not  vanish  when  j>-\-q-\/ —  1    is  put  for  X.     This  coefficient 
will  be  of  the  form  li-j-S-y/  —  1,  in  which  R  and  S  can  not  both  be  zero. 

When  p-{-q  \/ — 1  is  put  for  X,  we  have  represented  f(.r)  by  1'  +  ^-/  —  1- 
In  like  manner,  when  p^-q^/ — 1  — |—  A  is  put  for  X,  we  may  represont  the 
function  by  P'+Q'V — 1-      The  development  (2)  will  then  be 


P'  +  QV-l=(P  +  Qv'-l)  +  (K  +  >V-l)><",+  I'-nns 
/<"'+'.  /;'"+-',    ....    //". 

Now   //   is   quite   arbitrary;   we  may  give   to   it   any  sign   and   any  value  we 

please,  provided  only  it  come  under  the  general  form  <t-\-l>-\/ — 1.     Leaving 

the  absolute  valm  still  arbitrary,  we  may  therefore  replace  it  by  either  -|-fc 

I 

or  — /,-,  or  dj  ( —  1  )'"/,•;  and  thus  render//"1  either  positive  or  negative,  which' 

I 
ever  we  please,  whatever  be  tin-  value  of  m  ;  and  we  have  seen  that  ( —  i)™ 
comes  within  the  stipulated  form  (  \rt.  '.MO)-      llence  we  may  write  the  fore- 

going  development  thus,  the  Bign  ofjfc™  being  under  our  own  control: 


GENERAL  THEORY  OK  EQ.UAT10NS.  307 


P  +QV-l  =  (P+Q^-i)  +  (R+S^-l)t"+  terms  in 
/■•'+1,  /■"'+-,  ....  k". 

But  in  any  equation  of  this  kind  the  real  terms  in  one  member  are  together 
equal  to  those  in  the  other,  and  the  imaginary  terms  in  one  to  the  imaginary 
terms  in  the  other.     Consequently, 

P'=P-f  1U"'+  the  real  terms  in  km+\  km+2,  .  .  .  .  k"  : 
Q'  =  Q4-S/1:'"-f-  real  tonus  involving  powers  above  k'". 
Hence  the  square  of  the  modulus  of  P'+Q'V  —  1  is 
P's+Q/s=Ps+Qs+2(PR+QS)jfcm+  real  terms  in  k'"+\  k'"+\  .  .  k*\ 

Now  Jc  may  be  taken  so  small  that  the  sum  of  all  the  terms  after  P2-f-Qa 
may  take  the  same  sign  as  2(PR-}-QS)/»-"'  by  (239),  which  sign  we  can  always 
render  negative  whatever  PB--J-QS  may  be,  because,  as  observed  above,  km 
may  be  made  either  positive  or  negative,  as  we  please. 

Hence  we  can  always  render 

P/3+Q/a<Pa+Qs,  or  ^Fi+Q*<  VP2+Q2- 


In  other  words,  whatever  values  of  p  and  q,  in  the  expression  p-\-q  yf  — 1, 
cause  the  modulus  -y/P^+Q2  fo  exceed  zero,  other  values  exist  for  which  the 
modulus  will  become  smaller;  and,  consequently,  one  case  at  least  must  exist 
for  which  the  modulus,  and,  consequently,  the  expression  P-f-Q-y/  —  1,  must 
become  zero. 

This  conclusion  presumes,  however,  that  PR-f-QS  is  not  zero.  If  such 
jhould  be  tho  case,  then  our  having  chosen  the  form  of  h,  so  as  to  secure  a  com- 
mand over  the  sign  of  '2(PIl-f-QS),  will  have  been  unnecessary.  The  form 
must  then  be  so  chosen  that  a  command  may  be  secured  over  the  sign  of  the 
first  term  after  2(PR+QS)&m,  in  the  above  series,  for  P'2-f  Q'2,  which  does 
not  vanish,  when  the  preceding  conclusion  will  follow. 

242.  The  values  of  a  and  b  in  the  expression  a-\-b  y/ — 1,  which,  when  put 
for  x  in/(.r),  cause  that  polynomial  to  vanish,  can  never  be  infinite. 
We  may  write /(.r)  as  follows,  viz., 

n  v        „( -,   i  An— i  ,  An_2  N\ 

/(.r)=.r^l  +  —  +— +   ...^5 


or,  putting  P  +  Q-y7 — 1  for  what  f(x)  becomes,  when  p-\-qy/ —  1  is  substi- 
tuted for  .r,  we  have 

P+Qa/^1= 

/  An_!  An_2  N  \ 

Now  the  modulus  of  a  quotient  is  the  quotient  of  the  modulus  of  the  divi- 
dend by  the  modulus  of  the  divisor  (Lemma  II.).  In  each  of  the  dividends 
An_i,  An_2,  &c,  above,  the  modulus  is  finite  by  hypothesis.  Hence,  if  either 
p  or  q  be  infinite,  and,  consequently,  the  modulus  of  every  denominator  or 
divisor  also  infinite,  the  modulus  of  each  quotient  must  be  zero.  Hence,  in 
this  case,  each  of  the  above  fractions  must  itself  be  zero  (Lemma  III.),  and 
therefore  the  modulus  of  the  entire  quantity  within  the  parenthesis  simply  1 ; 
and  the  modulus  of  a  product  is  the  product  of  the  moduli  of  the  factors,  so 
that  the  modulus  of  the  preceding  product,  viz.,  -/P'  +  Q"'  is  tne  modulus  of 
(P~\-qV  —  !)"•  But  the  nth  power  of  j>-{-qV — 1  has  for  modulus  the  n,h 
oovyer  of  the  .modulus  o(p-\-q  V  —  1,  that  is,  the  nxb  power  of  Vl}2-{-<f  (Lemma 


.JU8  ALCiKHRA. 


IL,  Cor.),  which  is  infinite;  consequently,  VP'+^i"  must  be  infinite.  But 
when  p  +  7  V  —  1  is  a  root  of  the  equation  f(x)  =  0,  \/P"+Q2  is  zero.  Hence, 
in  this  case,  neither  p  nor  q  can  he  infinite. 

'243.  An  objection  may  be  brought  against  the  preceding  reasoning  that 
ought  not  to  be  concealed.  It  may  be  denied  that  the  modulus  of  the  product 
above  referred  to  is  simply  the  modulus  of  (p-\-q  V  — l)n  in  tho  case  of  j?  or  q 
infinite;  for  il  may  be  maintained  that  although  in  this  case  all  the  quantities 
within  the  parenthesis  after  the  1  become  zero,  yet  tho  combination  of  these 
with  {p>-\-<]  V  —  1)".  which  involves  infinite  quantities,  may  produce  quantities 
also  infinite  ;  and  thus  the  modulus  of  the  product  may  differ  from  tho  modu- 
lus of  (p+<7  v  — l)n  by  a  quantity  infinitely  great.  It  is  not  to  be  denied  that 
there  is  weight  in  this  objection.  But  it  is  not  difficult  to  see  that  although 
the  true  modulus  may  thus  differ  from  tho  modulus  of  {p-\-q-J — l)n  by  an 
infinite  quantity,  yet  the  modulus  of  (p+tfV  —  l)n,  involving  higher  powers 
than  enter  into  the  pait  neglected,  is  infinitely  greater  than  that  part.  This 
parti  therefore,  is  justly  regarded  as  nothing  in  comparison  to  the  part  pre- 
served, the  former  standing  in  relation  to  the  latter  as  a  finite  quantity  to  in- 
finity. 

But  the  proposition  may  be  established  somewhat  differently,  as  follows: 

Substituting  {]i-\-qV —  *)  f°r  x  mf(x)i  wo  have 

r+QV^i= 


(l>+?  V-l)n  + A„_1(jJ  +  g\/-l)r'-1+.-A1(i?+?-/-l)  +  N. 

Call  the  aggregate  of  all  these  terms  after  the  first  P'  +  Q'  -/ — 1  ;  then  it 
js  plain  that  tho  modulus  of  the  first  term,  that  is,  (  yfp*-\-q*)n,  must  infinitely 
exceed  the  modulus  VP'2+Q'2  °f  the  remaining  terms  whenever  p  or  q  is 
infinite,  because  in  this  latter  modulus  so  high  a  power  of  the  infinite  quantity 
p  or  q  can  not  enter  as  enters  into  the  former.  Now  the  modulus  of  the 
whole  expression,  that  is,  of  the  sum  of  (p-\-q  \f —  l)n  and  P'+Q'V  —  1-  is 
not  less  than  the  difference  of  the  moduli  of  these  quantities  themselves 
(Lemma  I.),  which  difference  is  infinite.  Nonce,  as  before,  "/PM-Q1  must 
be  infinite  whcn_p  or  q  is  infinite. 

PROPOSITION    IV. 

244.  Every  equation  containing  hut  one  unknown  quantity  has  as  many  roots 
as  there  are  units  in  the  highest  power  of  the  unknown  quantity. 

Let/(.r)  =  0  be  an  equation  of  tho  n,h  degree;  then  if  ax  be  a  root  of  this 
equation,  we  have,  by  Proposition  II. , 

(.r-^)/,(.r)=:/(.r)  =  0 (1) 

where/,  (.1)  represents  the  quotient  arising  from  the  division  of /(.»•)  byx — a,, 
nnd  will  be  a  polynomial,  arranged  according  to  the  powers  of  x,  ono  degree 
lower  than  the  given  polynomial  f(x).  The  equation  (1)  may  be  satisfied  by 
making  either  x — a ,  =0,  x=a , .  or  by  making/, (z)=0.  Bui  j\{.r)=0  must 
have  a  root,  as  aa  (see  Prop.  III.,  large  edition)  .'./^(x)  must  bo  divisible  by 
r— a3,  .•./,(.,)  =  (./■-,/,  )/,(./■). 

Substituting  this  value  of/,(x)  in  (1),  it  becomes 

Proceeding   r.  this  manner,  it'  s31  (/ ,,  CLt, a,  are  roots  of  the  BUCCi 


«   GENERAL  THEORY  OF  EQUATIONS. 

factors /2(x)=0,/3(x)  =  0 /„(.?•)  =  0,  the  degree  of  the  quotient  reducing 

l>y  one  each  time,  the  equation  will  assume  the  form 

(x—al){x—a2)(x—a3) (x— ge„)  =  0; 

and,  consequently,  there  are  as  many  roots  as  factors,  that  is,  as  units  in  the 
highest  power  of  .r,  the  unknown  quantity;  for  the  last  equation  will  be  veri- 
fied by  any  one  of  the  n  conditions, 

x=alt  x=a.2,  x=a3,  x=aA, ....  x=aa ; 

and  since  the  equation,  being  of  the  na  degree,  contains  n  of  these  factors  of 
the  1st  degree,  (x —  </,),  &c,  there  are  n  roots. 

Corollary  1.  When  one  root  of  an  equation  is  known,  the  depressed  equa- 
tion containing  the  remaining  roots  is  readily  found  by  synthetic  division. 

Corollary  2.  The  number  of  factors  of  the  2°  degree  in  an  equation  is  n(n  —  1 ) 
4jl  .  2;  of  the  3°,  n{n  —  l){n— 2)+l  .2.3,  and  so  on  (see  Art.  2031. 

EXAMPLES. 

(1)  One  root  of  the  equation  x1  — 25x3+60x— 36=0  is  3  ;  find  the  equation 
•ontaining  the  remaining  roots. 

1  _|_o    —25  -f  GO  —  30  (3 

3    +   9   —48+36 
1  +3—16+12. 
Henco  .r',+  3.r:  —  1  Or +12  =  0 

is  the  equation  containing  the  remaining  roots. 

(2)  Two  roots  of  the  equation  x'  — 12x3+48x2— 68x+15=0  are  3  and  5  ; 
find  the  quadratic  containing  the  remaining  roots. 

1  —12  +48—68+15  (3 

3  —27  +  03  —  15 

1  _  «j  +21—  5  (5 

5  —20—  5 


1—4+1 
...  x-—   4x+l=0 
•s  the  equation  containing  the  two  remaining  roots. 

(3)  One  root   of  the  cubic   equation  x3— 6x2+llx  —  6=0  is  1;   find  the 
■juadratic  containing  the  other  roots.  Ans.  x2 — 5.r+G=0. 

(4)  Two  roots  of  the  biquadratic  equation  Ax4 —  14a?  —  5x2  +  31x+6  =  0  are 
2  and  3;  find  the  reduced  equation.  Ans.  4x-  +  Gx+l=0. 

(5)  One  root  of  the  cubic  equation  x?-\-Zx'i— 16x+12=0  is  1  ;  find  the  re- 
maining roots.  Ans.  2  and  —  G. 

(6)  Two  roots  of  the  biquadratic  equation  x* —  Gx:,+24x — 16=0  are  2  and 
— 2  ;  find  the  other  two  roots.  Ans.  3+  ■s/b. 

proposition  v. 

245.    To  form  the  equation  whose  roots  are  a,,  a2,  a3,  a4, an. 

The  polynomial, /(x),  which  constitutes  the  first  member  of  the  equation 
required,  being  equal  to  the  continued  product  of  x— al,  x—a.j,  x—a3,  .  . 
r — a„,  by  the  last  proposition,  we  have 


*  There  can  be  no  other  factor  of  the  form  [x  —  a)  which  will  divide /x.  for,  if  there 
vrere,  it  must  divide  some  one  of  the  factors  (x  — a,),  (x—  a2),  Sec.     (See  note,  p.  83.) 


510 


ALOE  Bit  A. 


(x — a^x— aa)(x — a3) (x— au)=0; 

and  by  performing  the  multiplication  here  indicated,  we  have,  whex 

n=2,  x8 — ai 

— a. 

n=_3,  r3 — d) 
— a, 


x  -\-ala2=Q 

x  — ala:a:i=z0 


x-j-aiU;(73«4=0,  and  so  on. 


.r--\-alaz 

— a  3        -\-a2a3 
«  =  4,  .r' — «!     r3+aia2    z2 — CiOs^s 
—  a2        -f-«i«3        — aia.:a4 

— a4        +flia4        — a."/'-, 

+  "■'■'1 

By  continuing  the  multiplication  to  the  last,  the  equation  will  be  fonnil 
whose  roots  are  those  proposed  ;  and  from  what  has  been  done  we  learn  that 

(1)  The  coefficient  of  the  second  term  in  the  resulting  polynomial  will  be 
the  sum  of  all  the  roots  with  their  signs  changed. 

(2)  The  coefficient  of  the  third  terra  will  be  the  sum  of  the  products  of 
every  two  roots  with  their  signs  changed. 

(3)  The  coefficient  of  the  fourth  term  will  be  the  sum  of  the  products  of 
every  three  roots  with  their  signs  changed. 

(4)  The  coefficient  of  the  fifth  term  will  be  the  sum  of  the  products  of 
every  four  roots  with  their  signs  changed,  and  so  on  ;  the  last  or  absolitt' 
term  being  the  product  of  all  the  roots  with  their  signs  changed.* 

*  I.  The  generality  of  this  law  may  be  proved  as  follows  :  Let  us  suppose  it  to  hold 
good  for  the  product  of  n  binomial  factors,  we  shall  prove  that  it  will  for  the  product  of 
n-\-\  of  these.     Let 

xn—Alx"-1+Aixn--~,  &c,  4-A„ 

represent  the  product  of  n  binomial  factors,  in  which  A,  represents  the  sum  o1-f-fl.;+<i1 
-f-,  &.C.,  -\-an  of  the  n  second  terms  of  the  binomials,  A,,  the  sum  of  their  products  two  and 
two.  A3  the  sum  of  their  products  three  and  three,  and  so  on,  and  An  the  product  of  all  the 
n  second  terms  oia<2a3,  &c,  an. 

Introduce  now  a  new  factor  (x — an+1).  Performing  the  multiplication  of  the  above  poly- 
nomial by  this  new  factor, 

xn— A^-'+Aji"-'2— ,  &c,  ±  An 


x — a 


n  +  l 


-"+'- 


-A1xn+Aixn-1— ,  ice,  ±Aax 
-qn+1.T"+A1an+,x"-1-,  &c.  -T^ 


+« 


r"  +  '-A, 


'n+l 


*°+A« 

+Alr7n+l 


—A. 


-,  &c,  T-V'„+1 

Here  the  coefficient  of  the  second  t  is  composed  of  A  ,  the  sum  of  all  tho 

— "n+l  ' 

■econd  terms  of  the  n  binomials  (x — O,),  (.r — n.X  fca,  and  <rn  +  I,  the  second  term  of  the 

(n-4\-l)"'  binomial,  and  is,  therefore,  eqntQ  to  "he  sum  of  the  second  terma  of  the  h+1  Mno> 

+A 
minis.    Tin-  in,  ■nil-lent  ot  the  third  term  is  composed  of  A.,  the  ram  of  the  prod 

!     ^"n+l 

nets  of  the  w  second  terms  two  and  two,  and  Vn+i'  tnc  suiu  °f  •''  '"'  ',,|'"1S.  °acb 


GENERAL  THEORY  OE  EQUATIONS.  311 

Corollary  1. — If  the  coefficient  of  the  second  term  in  any  equation  be  0, 
that  is,  if  the  second  term  be  absent,  the  sum  of  the  positive  roots  is  equal  to 
the  sum  of  the  negative  roots. 

Corollary  2. — If  the  signs  of  the  terms  of  the  equation  be  all  positive,  the 
roots  will  be  ull  negative,  and  if  the  signs  be  alternately  positive  and  negative, 
the  roots  will  be  all  positive. 

Corollary  3. — Every  root  of  an  equation  is  a  divisor  of  the  last  or  absolute 
term. 

+A2 
multiplied  by  the  new  second  term  an_f.,  ;  hence  will  be  the  sum  of  the  products 

af  the  w-j-1  second  terms  two  .and  two- 

The  last  term  A„«Ir+1  is  the  product  of  An,  which  is  the  product  of  all  the  n  second  terms 
multiplied  by  the  new  second  term  «,l+1,  so  that  Anan,l  is  the  product  of  all  the  w-f-1  sec- 
ond terms. 

We  have  thus  proved  that  if  the  law  for  the  formation  of  the  coefficients  above  stated 
hold  good  for  a  certain  number  of  binomial  factors  ti,  it  will  hold  good  for  one  more,  or  /i-f-1. 
We  have  seen,  by  experiment,  that  it  holds  good  for  four,  it  therefore  holds  good  for  five  i 
if  for  five,  it  must  for  six,  and  so  on  ml  infinitum. 

II.  One  might  imagine,  at  first  view,  that  the  above  relations  would  make  known  the 
roots.  They  give  at  once  equations  into  which  these  roots  enter,  and  which  are  equal  in 
number  to  the  coefficients  of  the  equation  (excepting  the  coefficient  of  the  first  term,  which 
is  unity).  The  number  of  these  coefficients  is  equal  to  the  number  of  the  roots  of  the  equa- 
tion. Unfortunately,  when  we  seek  to  resolve  these  secondary  equations,  we  are  led  to  the 
very  equation  proposed,  so  that  no  progress  is  made. 

For  simplicity,  I  will  take  the  equation  of  the  3°  degree. 

*3+rV;-fQ,E+R=0 (1) 

Designating  the  three  roots  by  a,  b,  c,  we  have,  to  determine  the  roots,  the  three  re- 
lations 

P=— a~b— c 

d=<ib-\-ac-\-bc (2) 

R= — abc 

To  deduce  from  them  an  equation  which  contains  but  the  unknown  a,  the  most  simple 
mode  of  proceeding  is,  to  multiply  the  1°  by  a2,  the  2°  by  a,'  nd  add  tliem  to  the  3°. 
There  results 

Pa»4-Q.ra+R=— n3— a?b— <&c 

-\-a"b-\-a"c-\-abc 
y  — abc. 

Reducing,  and  transposing  the  term  — a3,  we  have 

rt3-f-Pa2+a«+R=0. 

The  unknown  quantities  b  and  c  are  thus  eliminated,  but  the  equation  resulting  is  of  th* 
same  degree  with  the  proposed.  From  the  symmetrical  form  of  the  relations  (2)  we  per 
ceive  that  the  elimination  of  a  and  b,  or  a  and  c,  would  have  been  attended  with  similai 
consequences. 

III.  To  find  the  sum  of  the  squares  of  the  roots  of  any  equation. 

-A^a+b+c.+l; 

.'.  A1«=a2+Z>'2+C2 . . .  -\-l2-\-<2{ab+ac-{-bc-{-  ....) 
=  sum  of  the  squares  -|-2Aj  ; 

.•.  sum  of  squares  ^A,2 — 2A  ,. 
To  find  the  sum  of  the  reciprocals  of  the  roots. 

(— I)""1  An_1=/>r- . . .  l+ac . . . l+ab . .  1+ . . 
;— l)nAn=abc...l; 

a^b^c  W  A„ 


312  ALGEBRA. 

Corollary  4. — In  any  equation,  when  the  roots  are  all  real,  and  Uie  last  or 
absolute  term  very  small  compared  with  the  coefficients  of  the  other  terms, 
then  will  the  roots  of  such  an  equation  be  also  very  small. 

EXAMPLES. 

(1)  Form  the  equation  whose  roots  are  2,  3,  5,  and  — G 

Here  we  have  simply  to  perform  the  multiplication  indicated  in  the  equa- 
tion 

(.r_2)(x-3)(x-5)(x+6)=0 , 
and  this  is  best  done  by  detached  coefficients  in  the  following  manner  : 

1—  2  (—3 
—   3+   G 
1—5+6  (—5 
_  5  +  25— 30 
1  —  10  +  31—30  (G 
0  G  — 60+18G— 180 

*  1  —  4 —29 + 156 —1 80 
...  x4—4^—29x2+15G.r— 1-^0  =  0  is  the  equation  sought. 

(2)  Form  the  equation  whose  roots  are  1,  2,  and  — 3. 

(3)  Form  the  equation  whose  roots  are  3,  — 4,  2+  \/3,  and  2 —  y/3. 

(4)  Form  the  equation  whoso  roots  are  3+  y/5,  3 —  -\/5,  and  — G. 

ANSWERS. 

(2)  x3  — 7x+6=0. 

(3)  x4— 3.V3— 15x*+49x— 12  =  0. 

(4)  r3— 32.r+24  =  0. 

PROPOSITION  VI. 

246.  No  equation  whose  coefficients  are  all  integers,  and  that  of  the  Itighesl 
power  of  the  unknown  quantity  unity,  can  have  a  fractional  root. 

If  possible,  let  the  equation 

.rn  +  An^'1x"-1+  ...  +A3^+Ao.r  +  Al.r+N  =  n, 
whose  coefficients  arc  all  integral,  havo  a  fractional  root,  expressed  in  its  low- 
est terms  by  j.     If  we  substitute  this  for  x,  and  multiply  the  resulting  equation 
bv  b"~l,  we  shall  have 

X  +  An_1an-'+ (-A3a36n-3+AaZ.n  »+No"  '  =  0. 

I  o 

In  this  polynomial,  every  term  after  the  firsl  is  integral :  hence  the  first  term 
must  be  integral  also.     But  j  being  a  fraction  in  its  lowest  terms,  y  must  also 

be  a  fraction  in  its  lowest  terms,  and  can  DOl  be  an  integral     (See  Note  to 

Art.  84.)     Therefore  the  jn-oposed  equation  can  not  have  a  fractional  root. 

PROPOSITION    VII. 

247.  If  the  signs  of  the  alternate  terms  in  an  <• [nation  he  changed,  the  signs 
of  all  die  roots  will  In  cluing,  </. 

Let  r*+AV   1  +  A.;       + V     tX+A    =0     •   ■    •   •   (1) 

be  u:  equation  ;  then,  changing  the  signs  of  the  alternate  terms,  we  have 

.rn_\iX.n-l_|_A,cn     •  _ ^A,     ,,.    |    £.„  —  <)    0-     .     (2) 

or  _,■"  +  . V   '  —  A  ,r"   :  + "i  \,   ,x±An=0   ...   (3) 


GENERAL  THEORY  OP  EQUATIONS.  313 

But  equations  (2)  and  (3)  are  identical,  for  the  sum  of  the  positive  terms  in 
each  is  equal  to  the  sum  of  the  negative  terms,  and  therefore  they  are  identi- 
cal. Now  if  a  be  a  root  of  equation  (1),  and  if  a  be  substituted  for  x  in  equa- 
tion (1)  and  —a  in  equation  (2),  if  n  be  an  even  number,  or  in  equation  (3) 
if  n  be  an  odd  number,  the  results  will  be  the  very  same  ;  and  since  the  for- 
mer is  verified  by  such  substitution,  a  being  a  root,  the  latter,  viz.,  equation 
(2)  or  (3),  as  the  case  may  be,  is  also  verified,  and  thereforo  —a  is  a  root  of 
the  identical  equations  (2)  and  (3). 

Corollary.— -If  the  signs  of  all  the  terms  are  changed,  the  signs  of  the  roots 
remain  unchanged. 

EXAMPLES. 

(1)  The  roots  of  the  equation  r>  —  6x3+llx— 6=0  are  1,  2,  3.  What  are 
the  roots  of  the  equation  r5-f-6x2+ll:r-f-6  =  0  ? 

Ans.  —1,  —2,  —3. 

(2)  The  roots  of  the  equation  .r1  —  6xa+ 2Ax— 16=0  are  2,  —2,  3±  y/5. 
Express  the  equation  whose  roots  are  2,  —2,  —3+  y/5,  and  —3—  y/5. 

Ans.  tf+Gx*— 24x— 16=0. 

PROPOSITION   VIII. 

248.  Surds  and  impossible  roots  enter  equations  by  pairs. 

Letxn+A1x"-1  +  A,.rn-2-l A„^1r+A^=0'  be  an  equation  having  a  root 

of  the  form  a-\-b  V  —  1  >  then  will  a  — by/  —I  be  also  a  root  of  the  equation  ; 
for,  let  a-\-b  y/  —  1  be  substituted  for  x  in  the  equation,  and  wo  have 

(rt  +  ftV^Tr+Ai^  +  fc  VZ:l)n_1+  ••••A,1_1(a-f&-/::^l)  +  An=0- 
Now,  by  expanding  the  several  terms  of  this  equation,  we  shall  have  a  series 
of  monomials,  all  of  which  will  be  real  except  the  odd  powers  of  6  V  —  1, 
which  will  be  imaginary.     Let  P  represent  the  sum  of  the  real  and  Q  yf  —  1 
the  sum  of  the  imaginary  terms  of  the  expanded  equation  ;  then 

P+QV^T=o, 

an  equation  which  can  exist  only  when  P  =  0  and  Q=0,  for  the  imaginary 
quantities  can  not  cancel  the  real  ones,  but  the  real  must  cancel  one  another, 
and  the  imaginary  one  another  separately. 

Again,  let  a  —  b  \/ — 1  be  substituted  for  x  in  the  proposed  equation;  then 
the  only  difference  in  the  expanded  result  will  be  in  the  signs  of  the  odd  powers 
of  by/  — 1,  and  the  collected  monomials,  by  the  previous  notation,  will  assume 


the  form  P— QV  —  1  but  we  have  seen  that  P=0  and  Q  =  0  ; 

...  P_QV=1=0, 

and  hence  a  —  by/  —  \  also  verifies  the  equation,  and  is  therefore  a  root. 

Such  roots  are  called  conjugate. 

In  a  similar  manner,  it  is  proved  that  if  a-\-  y/b  be  one  root  of  an  equation, 
a —  y/b  will  also  be  a  root  of  that  equation. 

Corollary  1. — An  equation  which  has  impossible  roots  is  divisible  by 

ja-— (a  +  b  V^T) \  \x  —  {a  —  b  y/ —1)\,  or  x3— 2ax-\-a*+b*, 

and,  therefore,  every  equation  may  be  resolved  info  rational  factors,  simple  or 
q  ladratic. 

(  'orollary  2. — All  the  roots  of  an  equation  of  an  even  degree  may  be  impos- 


314  ALGEBRA. 

sible,  but  if  they  are  not  all  impossible,  the  equation  must  have  at  least  two 
real  roots. 

Corollary  3. — The  producbof  every  pair  of  impossible  roots  being  of  the 
form  a"-\-b'2  is  positive;  and,  therefore,  the  absolute  term  of  an  equation 
whose  roots  are  all  impossible  must  be  positive. 

Corollary  4. — Every  equation  of  an  odd  degree  has  at  least  one  real  root, 
and  if  there  be  but  one,  that  root  must  necessarily  have  a  contrary  sign  to 
that  of  the  last  term. 

Corollary  5. — Every  equation  of  an  even  degree  whose  last  term  is  nega- 
tive has  at  least  two  real  roots,  and  if  there  be  but  two,  the  one  is  positive 
and  the  other  negative. 

PROPOSITION   IX. 

249.  The  m  roals  of  the  equation  X=0,  or 

a-m_l_pi.n,-i_^Qrn,-2_|_)  &c>  _0 [A-j 

jnust  he  of  the  form  a-\-b  V  — 1,  of  which  form  we  have  already  shown  (Art. 
241)  that  it  must  have  one. 

For,  let  a-f-i-y/  —  1  bo  the  root  whose  existence  is  demonstrated.  We 
know  (Prop.  II.)  that  the  polynomial  x'"-\-.  See.,  is  divisible  by  .r — (a -|- 0  V  —  1): 
but  when  we  effect  this  division,  the  quantities  a-\-b  -/ — 1,  P,  (I-  &c.,  can 
combine  only  by  addition,  by  subtraction,  and  by  multiplication  ;  then  the  co- 
efficients of  the  quotient  xm_1 -+-,  cVc,  will  still  bo  of  the  form  fl+i-/  — 1. 
Consequently,  the  equation  xm~1-^-,  &c,  will  also  have  al  leasl  one  root  of  the 
form  a'+fe'  V  —1 ;  dividing  x'"-'-^,  &c,  by  x—(a'-\-b'  V  — 1),  the  coefficient* 
of  the  quotient  xm-24-i  &C.,  will  be  still  of  the  same  form.  Continuing  to 
reason  thus,  it  is  evident  that  the  primitive  polynomial  X  will  be  divided  into 
m  factors  of  the  form  X— {a-\-b  \/  —  1),  and,  consequently,  the  roots  of  the 
equation  will  all  be  of  the  form  a-\-b  V  —  1. 

PROPOSITION   X. 

250.  The  roots  of  the  two  conjugate  equations, 

Y+z/=T=o (1) 

Y  — Z  V— 1=0 (2) 

will  be  conjugates  of  each  other. 

Letx=a4-6-v/  — 1  be  a  root  of  equation  ( I ),  and  Y'-J-Z'  ■/  —  1  the  quotient 
of  its  first  member,  by  x — a  —  b  V — 1,  we  have  the  identity 

(Y'+Z'v^K*— a— &/^i)=Y+Zvf"^I (3) 

Effecting  the  multiplication  in  the  1°  member,  we  find 

(r  — ,/)Y'  +  oZ'+[(r— «)Z'  — M"]-/3l 
( lhanging  now  in  tho  two  factors  Z'  into  —  /',  and  b  into  —  b,  we  see  that 
in  the  producl  the  part  which  does  not  contain   V  —  1   remains  the  same,  and 
thai  thai  which  does  contain  \/— !  only  changes  its  sign;  by  virtue  of  (3), 
therefore,  w  e  have 

(V_ZV^l)(,r- a-\-by/^l)  =  Y-Z  -/^T  .•••(!) 
Prom  whence  we  conclude  that  a—b^ —  1  is  a  root  of  (2);  that  is,  all  the  ro 
qf  (•>)  are  obtained  b\  changing  in  those  "!'  (l)  the  sign  of  ■/— 1.     The  real 
Mots,  according  to  this,  musl  bo  the  same  in  the  two  equatio 


GENERAL  THEORY  OF  EQUATIONS.  31b 

We  may  now  consider  the  following  beautiful  proposition  as  demonstrated 
from  the  foregoing. 

PROPOSITION  XI. 

An  algebraic  equation  which  has  real  coefficients  is  always  composed  of  as 
many  real  factors  of  the  1°  degree  as  it  has  real  roots,  and  of  as  many  real 
factors  of  the  2°  degree  as  it  lias  pairs  ofHmaginary  roots. 

DEPRESSION  OR  ELEVATION  OP  ROOTS  OF  EQUATIONS. 

PROPOSITION. 

251.  To  transform  an  equation  into  another  whose  roots  shall  he  the  roots  oj 
the  proposed  equation  increased  or  diminished  by  any  given  quantity. 

Let  a.Tn  +  A1.rn-14-A2a:n-2-j- An_!.r4- A„=0,  be  an  equation,  and  let  it 

be  required  to  transform  it  into  an  equation  whose  roots  shall  be  the  roots  of 
this  equation  diminished  by  r. 

This  transformation  might  lie  effected  by  substituting  y-\-r  for  x  in  the  pro- 
posed equation,  and  the  resulting  equation  in  y  would  be  that  required;  but 
this  operation  is  generally  veiy  tedious,  and  we  must  therefore  have  recoi 
to  some  more  simple  mode  of  forming  the  transformed  equation.  If  we  write 
y-\-r  for  x  in  the  proposed  equation,  it  will  obviously  be  an  equation  of  the 
very  same  dimensions,  and  its  form  will  evidently  be 

«r+Bi2/n_1  +  B^n_2+ BI1_17/4-Bn=0 (1)* 

in  which  Bi,  B3,  &c,  will  be  polynomials  involving  r.     But  y=x—r,  and  there- 
fore (1)  becomes 

a(z—r)-+B1(x— r)»-»+ Bn_1(a:-r)+Bn=0    .  .  (2) 

which,  when  developed,  must  lie  identical  with  the  proposed  equation;  for, 
since  y-\-r  was  substituted  for  x  in  the  proposed,  and  then  x — r  for  _y  in  (1) 
the  transformed  equation,  we  must  necessarily  have  reverted  to  the  original 
equation  ;  hence  we  have 
a(x—ry+'Qx(x— r)n~]  +  . .Bn_,(.r— r)-f  Bn=a.rn  +  A,a:n-1+  ..  A^x+A,,. 

*  It  will  be  of  the  same  form  with  the  development  in  the  note  to  (Art.  239).  We  give 
it  again  below,  arranged  according  to  the  powers  of  r  instead  of  y.  After  substituting  y-\-f 
for  r,  we  write  the  development  of  each  term  of  the  proposed  equation  in  a  horizontal  line  ; 
the  first  horizontal  line  is  the  development  of  a.rn,  the  second  of  A^x*— 1,  and  so  on. 

n  I          n-l                  ,  an{n — 1)„n-S  2  , 

ay*+any      r  -\ ^—  y     r         +... 

..„_!,.,  ,,     n-2      ,A](«— l)(?t— 2)  n_3   „ 

4-Aj^     -fA^H— \)y    H Y7z y    *"  +  •■• 

I     A       "-2    ,    A    ,  ^    n-3      ,  Ac(M— 2H«— 3)  „_,   „ 

.     +A;#      +A2(w— <2)y      r-\ — y      r  +  ... 


+A„. 
In  which  the  first  column  is  of  the  same  form  as  the  proposed  equation  ;   the  second 
column,  or  coefficient  of  r,  is  derived  from  the  first  by  multiplying  the  coefficient  of  each 
term  by  its  exponent,  and  diminishing  the  exponent  l>y  unity;  the  third  column,  or  coclfi 

cient  of ,  is  derived  from  the  second  in  a  similar  manner,  and  so  on. 

1.9 

If  we  designate  byf[x)  the  first  member  of  the  given  equation,  and  byf'(x)  the  first  de- 
rived function,  by/"(.c)  the  second  derived,  and  so  on,  we  shall  have 

fix)  f'"(x) 


316  ALGEBRA. 

Now,  if  we  divide  the  first  member  by  X — r,  every  term  will  evidently  be  divis- 
ible, except  the  last,  Bn,  which  will  be  the  remainder,  and  the  quotient  will  be 

a(z— rJ-'+B^ar— r)-*+ Bn_:(.r-r)+B„_i; 

and  since  tne  second  member  is  identical  with  the  fust,  the  very  same  quotient 
and  remainder  would  arise  by  dividing  this  second  member  also  by  x — r  ; 
hence  it  appears  that  if  the  first  member  of  the  original  equation  be  divided  by 
x  —  r,  the  remainder  will  be  the  last  or  absolute  term  of  the  sought  transformed 
equation. 

Again,  if  we  divide  the  quotient  thus  obtained,  viz., 

t/(.r_r)..-i  +  BI(.r-r)»~-  +  ....  Bn_2(r- r)  +  B„_, 
by  x — r,  the  remainder  will  obviously  be  B„_i,  the  coefficient  of  the  term  last 
but  one  in  the  transformed  equation;  and  thus,  by  successive  divisions  of  the 
polynomial  in  the  first  member  of  the  proposed  equation  by  X  —  r,  we  shall  ob- 
tain the  whole  of  the  coefficients  of  the  required  equation. 

RULE. 

Let  the  polynomial  in  the  first  member  ofHhe  proposed  equation  be  a  func- 
tion of  r,  and  r  the  quantity  by  which  the  roots  of  the  equation  are  to  be  di- 
minished or  increased  ;  then  divide  the  proposed  polynomial  by  x — r,  or  .r-f-'" 
according  as  the  roots  of  tho  proposed  equation  are  to  be  diminished  or  in- 
creased, and  the.  quotient  thus  obtained  by  the  same  divisor,  giving  a  second 
quotient,  which  divide  by  the  same  divisor,  and  so  on  till  the  division  termi- 
nates:  then  will  the  coefficients  of  the  transformed  equation,  beginning  with 
the  highest  power  of  the  unknown  quantity,  be  the  coefficient  of  the  highest 
jkower  of  the  unknown  quantity  in  the  proposed  equation,  and  the  re- 

mainders arising  from  the  successive  divisions  taken  in  a  reverse  order,  the 
first  remainder  being  the  last  or  absolute  term  in  the  required  transformed 
equation. 

Pjpte. — When  there  is  an  absent  term  in  tho  equation,  its  place  must  be 
supplied  with  a  cipher. 

EXAMPLES. 

(1)  Transform  the  equation  5-r4— 12r'-4-3.r2-|-4.r — 5=0  into  another  whose 
roots  shall  be  less  than  those  of  the  proposed  equation  by  2. 

X— 2)  5r>  —  12.r3+3.r2-f4.r— 5  (5.T3— 2x*— x+2 
5-r4  —  Id-'1 


— 2r3+3.r2 

—  2r3+4.r2 

— .r--j-4.r 

— a-2+2.r 

2x— 5 

2.1—4 

First 

remain 

—  1. 

der. 

r— 2)  5-r3  —  2r-  —  r+2 

(.',     4_e.r-fl5 

5a?— lOr* 

8xQ  —  X 

8.r2—  1 1:  -• 

1 5x 

+    2 

1 5x 

—30 

Second  remainder. 


GENERAL  THEORY  OF  EQUATIONS.  317 

x— 2)  5x2+8:r+15  (5.r+18 
5x2— 10.r 

18*+ 15 
18x— 36 

51.     Third  remainder 
x—2)  5x+18  (5 
5x— 10 


28.     Fourth  remainder. 
Therefore  the  transformed  equation  is 

53/4+28?/3+5l7/e+327/  — 1=0. 
This  laborious  operation  can  bo  avoided  by  Horner  s  Synthetic  Method  of 
division,  and  its  great  superiority  over  the  usual  method  will  be  at  once  ap- 
parent by  comparing  the  subsequent  elegant  process  with  the  work  above. 
Taking  the  same  example,  and  writing  the  modified  or  changed  term  of  the 
divisor  x — 2  on  the  right  hand  instead  of  the  left,  the  whole  of  the  work  will 
be  thus  arranged : 

5  —  12  +   3  +4—5  (2 
10   —  4   —  2       4 

~H  ~—[.-.  Bi=—  1 
30 


■&.' 


2 

—  1 

10 

1G 

8 

15 

10 

36 

18 

51 

10 

32  .-.  B3=32 
.  .*.  B3=51 

28  .-.  B1==28 

...  <jyi^.23y*-{-!>iy'2-\- 32;/  — 1=0  is  the  required  equation,  as  before. 
(2)   Transform  the  equation   5y4+28i)/,+51i/2+322/ — 1=0   into  anothei 
having  its  roots  greater  by  2  than  those  of  the  proposed  equation. 

5+28  +      51   _|_32   _i  (_2 


—  10 

—  36  - 

-30 
2 

—  4 

18 

15 

—5 

—  10 

—  16 

2 

4 

8 

—  1 

—  10 

4 

—  2 

3 

—10 

—  12 
...  5x4 — 12a3+3:i?+4a: — 5=0  is  the  sought  equation,  which,  from  the  trans- 
formations we  have  made,  must  be  the  original  equation  in  Example  1. 

(3)  Find  the  equation  whose  roots  are  less  by  1-7  than  those  of  the  equation 

z8— 2zs+3z— 4=0. 
1—2  +3  —4  (1 


1 

—  1 

—1 

o 

1 

0 

~~ 0 

2 

1 

1 

3  13  ALGEBRA. 

Now  we  know  the  equation  whose  roots  are  less  by  1  than  thjse  of  the 
iriven  equation*  it  is  3?-\-3?-\-2x  —  2=0  ;  and  by  a  similar  process  for  -7,  re- 
m. -inhering  the  localities  of  the  decimals,  we  have  the   required   equation  ; 

tlaiii  : 

1  +  1       +2        —2  (-7 
.7        1-19       2-233 
•233 


1-7 

3-19 

7 

1-68 

2-4 

4-87 

7 

3-1 

.-.  3/3+3-l2/2+4-873/+-233  =  0  is  the  required  equation. 

This  latter  operation  can  be  continued  from  the  former  without  arranging 
the  coefficients  anew  in  a  horizontal  line,  recourse  being  had  to  this  second 
operation  merely  to  show  the  several  steps  in  tho  transformation,  and  to  point 
out  the  equations  at  each  step  of  the  successive  diminutions  of  the  roots. 
Combining  these  two  operations,  then,  we  have  the  subsequent  arrange- 
ment. 


1—2 

+  3 

—4  (1-7 

1 
—  1 

— 1 

~~ 2 

2 
^2 

1 

~0 

0 
2 

2-233 
•233 

1 

1-19 

1-7 

3-19 

•7 

1-68 

2-4 

4-87 

•7 

3-1 

1—2 

1-7 

+  3 
—   -51 

—4  (1-7 
4-233 

—  -3 

2-49 

•233 

1-7 

2-38 

1-4 

4-87 

1-7 

3-1 

We  have  then  the  same  resulting  equation  as  before,  and  in  the  latter  of 
these  we  have  used  1-7  at  once.  It  is  always  better,  however,  to  reduce 
continuously  as  in  the  former,  to  avoid  mistakes  incident  to  the  multiplier  1-7. 

(1)  Find  the  equation  whose  roots  shall  be  loss  by  1  than  those  of  the 
equation 

;r>— 7.r +7  =  0. 

(5)  Find  the  equation  whoso  roots  shall  he  less  by  3  than  the  roots  of  the 
equation 

x*  —  3.r»  — 1  -,.r-+4'.t.r— 12  =  0. 
mid  transform  the  resulting  equation  into  another  whose  roots  shall  bo  groater 
by   I. 


GENERAL  THEORY  OF  EQUATIONS.  319 

(6)  Give  the  equation  whose  roots  shall  be  less  by  10  than  the  roots  of  the 

equation 

r«_j_  ori_|_  3.C2_|_  4x— 12340 =0. 

(7)  Give  the  equation  whoso   roots  shall  be  less  by  2  than  those  of  the 
equation 

x*  -J-  2xa — 6x2  —  1  0x+  8 = 0 . 

(8)  Give  the  equation  whose  roots  shall  each  be  less  by  |  than  the  roots  oi 
the  equation 

2x*— G.r,+4.r2— 2.r+ 1=0. 

ANSWKRS. 

(4)  2/3 -|_ 3jy2 — 4t/ -j- 1  =  0 whence  x=y-\-   1 

(5)  y»_|_ 9y-i_|_i2j/3  —  14t/  =  0 whence  x=7/4-   3 

and  z4—7z3+66z  —  72=0 whence  x=z—   1 

(G)  i/<4-427/34-6G37/3+4GG4-y  =  0 whence  x=y  + 10 

(7)  7/54-10/y'4-42?/:,4-86?/2 4-70^/4-12  =  0 whence  x =y+  2 

(8)  2y4— Sy*— 2?/2— ^+8  =  ° whence  x=y-\-  i 

PROPOSITION 

252.  If  the  real  roots  of  an  equation,  taken  in  the  order  of  their  magnitudes,  be 

^11   ^2'   ^3'   ^41    ^5^ • 

where  a,?'s  the  greatest,  a2  tfie  we.r<,  and  so  on  ;  then  if  a  series  of  numbers, 

ft,,  fto,  ft-j,  ft4,  fts, 

sn  which  b,  is  greater  than  a,,  b2  a  number  between  a,  a«(/  a2,  b3  a  number 
between  a3  ««<Z  a3,  and  so  on,  be  substituted  for  x  in  the  proposed  equation, 
results  ivill  be  aMernatcly  positive  and  negative. 
The  polynomial  in  the  first  member  of  the  proposed  equation  is  the  product 
u!'  the  simple  factors 

{x— at)(x — aa)(x — a;i)(x— a4) 

and  quadratic  factors,  involving  the  imaginary  roots;  but  the  quadratic  factors 
have  always  a  positive  value  for  every  real  value  of  x  (Art.  248,  Cor.  3) ;  there- 
fore wo  may  omit  these  positive  factors ;  and  substituting  for  x  the  proposed 
series  of  values,  ft,,  b2,  ft3,  &c,  we  have  these  results: 

(ft,—  a,)(ft,—  a,.)(ft,—  a3)(bl—  aA)  ....  =  +  .  +  •  +  ■+ =  + 

(h.,-al)(b2-a2){b3-a3)(ba-aA)....  =_.  +  .  +  .+ =- 

{b:t—nt)(h,—a.,){b.,—a3){b3—aA)  ....=—.  —  .4-.+ =  + 

V>4—<h)(l>i—<i:){bA—a3){bA-aA)  ....=  —  . .-f- =  — 

&c.  &c.  6cc. 

Corollary  1. — If  two  numbers  be  successively  substituted  for  X  in  any  equa 
Hon,  ami  give  results  with  different  signs,  then  between  these  numbers  there 
must  be  one,  three,  five,  or  some  odd  number  of  roots. 

Corollary  2. — If  the  results  of  the  substitution  in  corollary  1  are  affected 
with  like  signs,  then  between  these  numbers  there  must  be  two,  four,  or  some 
even  number  of  roots,  or  no  root  between  these  numbers. 

( 'orollary  3. — If  any  quantity  q,  and  every  quantity  greater  than  q,  renders 
the  result  positive,  then  q  is  greater  than  the  greatest  root  of  the  equation. 

Corollary  4. — Hence,  if  the  signs  of  the  alternate  terms  be  changed,  and  if 
p,  and  every  quantity  greater  than  p,  renders  the  result  positive,  then  — p  is 
less  than  the  least  root. 


320 


ALGEBHA 


i  \  ample. 
Find  the  initial  figure  in  one  of  the  roots  of  the  equation 

X3 — i. ,-•_(;.,•  -|-8  =  0. 
Here  one  value  of  x  does  not  differ  greatly  from  unity,  for  the  value  of  the 
given  polynomial,  when  x=l,  is  — 1,  and  when  r=-9,  it  is  found  thus; 

1_4_G      4-8  (-9 
•9—2-79—7911 


—31— 8-79+  -089. 
The  value,  therefore,  when  .r=-9  is  (Art.  251)  -089.     Hence  the  former 
alue  being  negative,  and  the  latter  positive,  the  initial  figure  of  one  root  is  -9. 

PROPOSITION. 

253.   Given  an  equation  of  the  n"'  degree  to  determine  another  of  the  (n  — 
lefrrcc,  such  that  the  real  roots  of  the  former  shall  separate  those  of  the  latte 


Let  a.,  a 


a-,,  a 


■21  "3' 


|1 


aa  be  the  roots  taken  in  order  of  the  equation 


xn  4-  A ,  .r"-]  4-  A  2xn-2  4 An_!.r  4-  An  =  0  ; 

then  diminishing  the  roots  of  this  equation  by  r  (Art.  251),  we  have  the  U 
lowing  process,  viz. : 

14-A,4-  A24- An_24-  An_,4-  An  (r 


r 
r 


rB, 
rC, 


rBn_3     rBn 


B„_3 
rC„_3 

Cn_2 


Bn_l 

rCn-? 

C„_! 


rBn_x 
"b7~ 


Whence 

Cn_1=A1_,+  r  B0_2-f  r  C„_3  • 

=An_!4-  r(An_24-  r  Bn_3)4-  r  (An_,  +r  Bn_34-rCn_3) 
=An_!4-2r  A„_s+2r«  Bn_3  +  r"  CB_3 

=  A„_i  +  2r  An_24-2r-(A„_3  +  r  Bn_4)4-r-(An_34-rB„_,4-rC0_^) 
=A»_i+2r  An_24-3r2  An_3  4-3r>  Bn_,  4-r3  Cn_, 


=An_14-2r  An_24-3r2An_34- (n— l)i*-*Ai+ftr 


or 


( 


Cn_1=nr"-'4-(»— l)A1r"--4-(/i  — 2)A.:r"-34-  .  .  .  2An_Sr+AB-I 
Again,  the  roots  of  the  transformed  equation  will  evidently  be 
fli — r,  a2—r,  a3—r,  a^—r,  ....  an— r, 
and  as  we  have  found  the  coefficient,  C„-n  of  the  last  term  but  one  in  th 
transformed  equation,  by  one  process,  we  shall  now  find  the  same  coefficien 
Co-d  by  another  process  (Prop.  V.,  p.  3(fe) ;  il  is  the  product  of  every  (n  —  1) 
roots  of  the  equation  (1)  with  their  signs  changed;  hence  wo  have 

Cn_l  =  (r—al){r—a:){r  —  ai) to  (n— 1)  factors  ' 

+  (r— ai)(r— th)[r— a<) to  (n—  1)  factors 

4-(r— ai)(r— ".)('•— oi) to  (n— 1)  factors 


4-(r  —  a:){r  — flj)(r— a*) to  (n— 1)  (actors 

Now  these  two  expressions  which  we  have  obtained  for  Cn_i  are  equal  to 


GENERAL  THEORY  OF  EQUATIONS.  321 


' 

N 

,< 

. 

320 


ALGKBHA 


5-       " 


- 


' 


GENERAL  THEORY  OP  EQUATIONS.  321 

one  another,  and,  therefore,  whatever  changes  arise  by  substitution  in  the 
one,  the  same  changes  will  be  produced,  by  a  like  substitution,  in  the  other; 
nence,  substituting  au  a2,  a3,  &c,  successively  for  r  in  the  second  member  of 
equation  (2),  we  have  these  results  : 

(«i  —  a2)(ai— fi3)(ai—  a4) =  +  .  +  .+ =  + 

(a2— ai)(«2— a3)(a2  — a4) =—.  +  .-{- =  — 

(az—ai){a3—a2){a3—a4) =_._._|_ =4. 

&c.  &c.  &c. 

But  when  a  series  of  quantities,  au  a2,  as,  at,  &c,  are  substituted  for  the 
unknown  quantity  in  any  equation,  and  give  results  which  are  alternately  -\- 
and  — ,  then,  by  Art.  252,  these  quantities,  taken  in  order,  are  situated  in  the 
successive  intervals  of  the  real  roots  of  the  proposed  equation;  hence,  making 
C„_ir=0,  and  changing  r  into  .r,  wo  have  from  equation  (1) 

nxn-l-\-(n  —  l)Alxn-n'-\-{n—2)Aix"-3-\ 2An_2.r+An_1  =  0  ...  (3) 

an  equation  whose  roots,  therefore,  separate  those  of  the  original  equation 
x»4.A1a^-1+Asa;,l-3+  . . ..  An_,.r-f  An=0, 

and  the  manner  of  deriving  it  from  the  proposed  equation  is  to  multiply  each 
term  of  the  proposed  equation  by  the  exponent  of  x,  and  to  diminish  the  ex- 
ponent one.  It  is  identical  with  the  second  column  of  the  development  in 
the  note  to  Article  251.  It  is  known  by  the  name  of  the  derived  equa- 
tion. 

Let  au  a2,  a3,  a4,  &c,  be  the  roots  of  the  proposed  equation,  and  bt,  b2,  b3, 
&c,  those  of  the  derived  equation  (3),  ranged  in  the  order  of  magnitude  ;  then 
the  roots  of  both  the  given,  and  the  derived  equation  will  be  represented  in 
order  of  magnitude  by  tho  following  arrangement,  viz.  : 

ax,  6M  a2,  b2,  a3,  63,  a4,  64,  cr5,  b5,  &c.  .  . 

Corollary  1. — If  a2=au  then  r — a^  will  be  found  as  a  factor  in  each  of  the 
groups  of  factors  in  equation  (2),  which  has  been  shown  to  be  the  separating 
equation  (3),  and,  therefore,  the  separating  equation  and  the  original  equation 
will  obviously  have  a  common  measure  of  the  form  .r  —  «[. 

Corollary  2. — If  a3=a2=a1,  then  (r — «i)(r — a^)  will  occur  as  a  common 
factor  in  each  group  of  factors  in  (2) ;  that  is,  the  separating  equation  (3)  is  divis- 
ible by  (x — ax)~ ;  and,  therefore,  the  proposed  equation  and  the  separating  equa- 
tion have  a  common  measure  of  the  form  (x — ai)". 

Corollary*?,. — If  the  proposed  equation  have  also  a4=a5,  then  it  will  have  a 
common  measure  with  the  separating  equation  of  the  form  (x — a^  (r — a4), 
and  so  on. 

Scholium. — When,  therefore,  we  wish  to  ascertain  whether  a  proposed 
equation  has  equal  roots,  we  must  first  find  the  separating  equation,  and  then  find 
the  greatest  common  measure  of  the  polynomials  constituting  the  first  mem- 
bers of  these  two  equatious.  If  the  greatest  common  measure  be  of  the  form 
(x—aiy  (x—a2y{x—a3)r 

then  the  proposed  equation  will  have  (p-f-1)  roots  =al,  (<7+l)  roots  =a2 
(r-j-1)  roots  =a3,  &c.  The  equation  may  then  be  depressed  to  another  of 
lower  dimensions,  by  dividing  it  by  the  difference  between  .r  and  the  repeated 
root  raised  to  a  power  of  the  degree  expressed  by  the  number  of  times  it  ia 
repeated. 

X 


322  ALGEBRA- 

IC IMPIiES. 

Fiud  the  equal  roots  of  tho  equation 

.r_f-5.i*_j_Gx5— 6x*—  15s»— 3.c:+8x+4  =  0 (1) 

The  derived  polynomial  is 

71*4-30.^ -r-30.r<— 24.r!— 4o.f3— 6.r-f  8 (S 

and  the  common  divisor  of  (1)  and  (2) 

r,+3.r3+2-— 3.r  —  2 (3, 

The  values  of  x,  found  by  putting  this  equal  to  zero,  would  be  the  repeated 
roots  of  the  proposed  equation.  This  itself  will  be  found  to  have  equal  roots, 
for  its  derived  is 

4x3+9£3+2z— 3, 
and  their  common  divisor 

r+1. 
Hence,  by  the  rule, 

(z+1)9 (4) 

is  a  factor  of  (3),  and 

,(.t+l)' 
u  factor  of  the  proposed. 

Dividing  (3)  by  (4),  the  quotient  is 

.r-  +  .r— 2, 
which,  put  equal  to  zero,  gives 

x=l,  or  — 2. 
Hence  (3)  may  be  put  under  the  form 

(s+1)9  (s-1)  (*+2), 

and  by  the  rule  in  tho  above  scholium  the  given  equation  may  be  put  under 
the  form 

(.r+l)3  (*_!);  (a:+2)», 
so  that  in  the  proposed  equation  there  are  three  roots  equal  to  — 1,  two  to 
-f-1,  and  two  to  — 2. 

(2)  x3— 3a-x— 2a3=0. 
By  the  process  above  it  may  be  transformed  into 

{x+a)*  (x-2a)=0, 
so  that  the  three  roots  are  two  equal  to  — a,  and  the  third  2a. 

(3)  x8— 12x7+53.r6— 92r'i— 9.r«+212.i'3  — 153xs— 108.r  -f-108  =  0 
decomposes  into 

(.r— 1)  (x— 2)-  (.r+1)-  (.i  — 3)3=0. 
254.  The  most  satisfactory  and  unfailing  criterion  for  the  determination  of 
the  number  of  imaginary  roots  in  any  equation  is  famished  by  tho  admirable 
theorem  of  Sturm,  which  gives  the  precise  number  of  real  roots,  and,  conse- 
quently, the  exact  number  of  imaginary  ones,  since  both  the  real  ami  imagi- 
nary roots  are  together  equal  to  tho  number  denoted  by  the  degree  of  the 
proposed  equation. 

PROPOSITION. 

To  find  the  numher  of  real  and  imaginary  roots  in  any  proposed  equation. 

Tho  acknowledged  difficulty  which  has  hitherto  been  experienced  in  the 
important  problem  of  tho  separation  of  the  real  and  imaginary  roots  of  any 
proposed  equation  is  now  completely  removed  by  the  recent  valuable  re- 
searches of  the  celebrated  Sturm;  and  we  shall  now  give  the  demonstration 
of  the  theorem  by  which  this  desirable  object  has  been  so  fully  accomj 


THEOREM  OF  STURM.  323 

ed.  nearly  as  given  by  the  author  himself,  deeming  it  far  more*  satisfactory  thun 
any  other  version  which  we  have  seen. 

THEOREM  OK  STURM. 

I.  Let  N.r'"+P.im-1  +  Q.rm-=+ +Tar-j-U=0 

be  a  numerical  equation  of  any  degree  whatever,  of  which  it  is  proposed  to 
determine  all  the  real  roots. 

\Vre  begin  by  performing  upon  this  equation  the  operation  which  serves  to  de- 
ne whether  or  not  it  has  equal  roots  (Art.  253,  Sch.),  in  a  manner  which 
we  proceed  to  point  out.  If  V  designate  the  entire  function  N.cm-f-P*m-1  +  » 
&c,  and  Vi  its  derived  function  (which  is  formed  by  multiplying  each  term 
of  V  by  the  exponent  of  ;r  in  this  term,  and  diminishing  that  exponent  by  uni- 
ty), wo  must  seek  for  the  greatest  common  divisor  of  the  two  polynomials  V 
and  Vj.  Divide,  at  first,  V  by  V,,  and  when  a  remainder  is  obtained  of  a 
degree  inferior  to  that  of  the  divisor  Vj,  change  the  signs  of  all  the  terms  of 
this  remainder  (the  signs  -}-  into  —  and  —  into  -f-).  Designate  by  V3  what 
this  remainder  becomes  after  the  change  of  signs.  Divide  in  the  same  man- 
ner V,  by  Vn,  and,  after  having  changed  the  signs  of  the  remainder,  it  becomes 
a  new  polynomial  V3,  of  a  degree  inferior  to  that  of  V2.  The  division  of 
y  \  ;,  conducts,  in  the  same  manner,  to  a  function  /.,,  which  will  be  the 
remainder  resulting  from  this  division  after  having  changed  the  signs.  This 
series  of  divisions  is  to  be  continued,  taking  care  to  change  the  signs  of  the 
terms  of  each  remainder.  This  change  of  signs,  which  would  be  useless  if 
our  object  was  to  find  the  greatest  common  divisor  of  the  polynomials  V  and 
V,,  is  necessary  in  the  theory  about  to  be  explained.  As  the  degrees  of  the 
successive  remainders  go  on  diminishing,  we  arrive  finally  either  at  a  numeri- 
cal remainder  independent  of  x,  and  differing  from  zero,  or  at  a  remainder  a 
function  of  x,  which  exactly  divides  the  preceding  remainder. 

We  shall  examine  these  two  cases  separately. 

II.  Suppose,  in  the  first  place,  that,  after  a  certain  number  of  divisions,  we 
arrive  at  a  numerical  remainder,  which  may  be  represented  by  Vr. 

[n  this  case  wo  know  that  the  equation  V  =  0  has  no  equal  roots,  since  the 
polynomials  V  and  V,  have  no  common  divisor  function  of  x.     Representing  by 

Q,,  Q2 Qr-i,  the  quotients  given  by  the  successive  divisions,  which  leave 

for  remainders  — Vs,  — Vg —  Vr,  we  have  this  series  of  equalities  : 

V  =V,Ql-V.2 

VI  =  V:()..-V3 
V:=V3Q3-V4 (1) 

•  *       • 

Vrl,=Vr^IQr_1-Vr. 

Thus  much  being  premised,  the  consideration  of  this  system  of  functions 

V,  Vn  V3 Vr  furnishes  a  sure  and  easy  means  of  knowing  how  many  real 

oots  the  equation  V=0  has  comprehended  between  two  numbers  A  and  B  of 
any  magnitude  or  signs  whatever,  B  being  greater  than  A.  The  following  is 
the  rule  which  attains  this  object : 

Substitute  in  place  of  x  the  number  A  in  all  the  functions  V,  V,,  V3 

Vr_!,  Vr,  then  write  in  order,  in  one  line,  the  signs  of  the  results,  and  count 
the  number  of  variations  which  are  found  in  this  succession  of  signs.  Write, 
in  the  same  manner,  the  succession  of  signs  which  these  same  functions  take 
by  the  substitution  of  the  other   number  B,  and  ccmnt  the  number  of  variation* 


324  ALGEBRA. 

which  are  found  tn  this  second  succession.  The  number  of  varia  ons  which  it 
has  less  than  the  first  will  be  the  number  of  real  roots  of  the  equation  V  =  0 
comprehended  between  the  numbers  A  and  B.  If  the  second  succession  has  a.« 
many  variations  as  the  first,  the  equation  V=0  has  no  real  root  between  A 
and  B. 

III.  We  shall  demonstrate  this  theorem  by  examining  how  the  number  of 
variations  formed  by  the  signs  of  the  functions  V,  V,,  V; .  .  .  Vn  for  any  one 
value  whatever  of  x,  can  change,  when  x  passes  through  different  states  of 
magnitude. 

Whatever  may  be  the  signs  of  these  functions  for  one  determinate  value  of 
c,  when  x  increases'by  insensible  degrees  to  beyond  this  .value,  there  can  take 
place  no  change  of  signs  in  this  succession  of  signs,  unless  one  of  the  functions. 
V,  V,  ...,  changes  sign,  and,  consequently  (155,  note  3°),  becomes  zero 
There  are  then  two  cases  to  examine,  according  as  the  function  which  van- 
ishes is  the  first,  V,  or  some  one  of  the  other  functions,  V,,  V2  .  .  .  Vr_,,  in- 
termediate between  V  and  Vr :  the  last,  Vr,  can  not  change  sign,  since  it  is  a 
number  positive  or  negative. 

IV.  Let  us  see  first  what  alteration  the  succession  of  signs  experiences  when 
r,  in  increasing  in  a  continuous  manner,  attains  and  passes  by  a  value  which 
destroys  the  first  function  V.  Designate  this  value  by  c.  The  function  V, 
derived  from  V,  can  not  be  zero  at  the  same  time  with  V  for  x=c,  because 
by  the  hypothesis  the  equation  V=0  has  not  equal  roots.  We  see,  besides, 
by  the  equations  (1),  without  falling  back  upon  the  theory  of  equal  roots,  that 
if  the  functions  V  and  Vi  were  zero  for  x=c,  all  the  other  functions,  Va,  V3 
. . .,  and,  finally,  Vr,  would  be  zero  at  the  same  time ;  but,  on  the  contrary.  Vr 
is  by  hypothesis  a  number  different  from  zero.  V,  has  then  for  x=c  a  value 
dilferent  from  zero,  positive  or  negative. 

Let  us  consider  values  of  x  very  little  different  from  c.  If  in  designating  by 
u  a  positive  quantity  as  small  as  we  please,  we  make  by  turns  x=c — u  and 
z=c-\-u,  the  function  V,  will  have  for  these  two  values  of  x  the  same  sign 
that  it  has  for  .r=c  ;  because  we  can  take  u  sufficiently  small,  to  insure  that  V  , 
shall  have  for  these  two  values  of  x  the  same  sign  that  it  has  for  x=c  ;  since 
we  can  take  u  so  small  that  Vt  will  not  vanish,  and  not  change  sign,  while  x 
increases  from  the  value  c — u  to  c-\-u.* 

We  must  now  determine  the  sign  of  V  for  u -=c-\-u.  Designate  for  a  mo- 
ment V  by  f(v),  V,  by /'(a"),  and  the  other  derived  functions  of  V  by/"(.r), 
f'"(x)  .  .  .  • ,/"'(')'  which  are  not  to  bo  confounded  with  Va,  V3,  &C.,  these 
latter  not  being  derived  functions.  When  we  make  x=c-\-u,  V  becomes 
f(c-\-u),  and  we  have  (see  note  to  Prop. III.,  Art.  239,  or  Art.  251) 

/(c+tt)===/(c)+/'(c)u+Y^8+n^«3+.&c.; 
r^ne^,  observing  that/(r)  is  zero,  and  that /'(c)  is  not, 

f(c+u)=u[f'(c)+f^»+{-^>S+-]- 
We  see  from  tl's  expression  of/(c-|-u),  that  in  attributing  to  u  very  small 


"  The  deft  cat;  point  on  whit  li  the  theorem  hinges  is  the  one  stnted  here.  Let  it  he  dis- 
tinctly seen  that  sHic*  V,  can  not  be  zero  at  the  6amc  time  with  V  when  .r  — <-,  therefore, 
however  little  c  maj  dt£er  from  a  value  which  reduce*  V,  to  zero,  u  may  be  taken  smaller 
than  this  difference 


THEOREM  OP  STURM.  325 

positive  values,  f{e-\-u)  will  have  the  same  sign  as/'.   \*  and,  consequently, 
J  (c-f-w)  will  have  also  the  same  sign  &sf'(c-\-u),  since  ''(c-j-w)  has  the  same 
sign  as /'(c).     Thus,  V  has  the  same  sign  as  V,  for  x=c-\-u. 
By  changing  u  into  — u  in  the  preceding  formula,  we  have 

/(c_M)-=_M[/'(c)_^i)t4+)  &c.] 

A.nd  we  perceive,  in  the  same  manner,  that  f(c — u)  has  a  sign  contrary  to 
that  of /'(c),  from  whence  it  follows  that  for  x=c — u  the  sign  of  V  is  contrary 
to  that  of  V , . 

Then,  if  the  sign  of /'(c)  or  of  Vt,  for  x=c,  is  -f-t  ^ie  s'gn  °f  V  W'U  be  4" 
for  x=c-T-u,  and  —  for  x=c — u.     If,  on  the  contrary,  the  sign  of  V!  is  - 
for  ar=c,  that  of  V  will  be  —  for  x=c-\-u,  and  -|-  for  x=c — u.    Besides,  V, 
has  for  x=c-\-u  and  for  x=c — u  the  same  sign  as  it  has  for  x=zc. 
These  results  are  indicated  in  the  following  table : 

V  V,  V  V, 

cx=c—u, (-,  -| , 

For                                <  x=c,           0    +,  or  else   0    — , 
{x—c-\-u,    +  +,  • 

Thus,  when  the  function  V  vanishes,  the  sign  of  V  forms  with  the  sign  ol 
V  ,  a  variation,  before  x  attains  the  value  c,  which  reduces  V  to  zero,  and  this 
variation  is  changed  into  a  permanence  after  x  passes  this  value. 

As  to  the  other  functions,  V2,  V3,  &c,  each  will  have,  as  V,,  either  for 
.r=c-|-w  or  for  x=c — u,  the  same  sign  that  it  has  for  x-=c,  that  is,  if  none  of 
them  vanish  for  x=c  at  the  same  time  with  V. 

The  succession  of  the  signs  of  the  functions  V,  V1(  V2  ...  V„  loses  then  a 
variation,  when  x,  going  on  increasing,  passes  over  a  value  c,  which  reduces 
the  first  function  V  to  zero  without  destroying  any  of  the  other  functions,  V,, 
V 21  &c.  It  is  necessary  now  to  examine  what  happens  when  one  of  these 
functions  vanishes. 

V.  Let  there  be  a  function,  V„,  intermediate  between  V  and  Vr,  which  is 
lestroyed  when  x  becomes  equal  to  b.  This  value  of  x  can  not  reduce  to  zero 
cither  the  function  V„_i,  which  precedes  immediately  Vn,  or  the  function 
\'„+i,  which  follows  Vn.  Indeed,  we  have  between  the  three  functions  Vn_!, 
\r„,  Vn+1,  the  following  equation,  which  is  one  of  the  equations  (1). 

Vn_1  =  VnQ,1-Vn+I. 

It  proves  that  if  the  two  consecutive  functions,  Vn_H  Vn,  were  zero  for  the 
same  value  of  x,  V„+1  would  be  zero  at  the  same  time ;  and  as  we  have  also 

we  should  have,  again,  V11+n=0,  and  so  on,  so  that  we  should  have  finally 
Vr=0,  which  is  contrary  to  the  hypothesis. 

The  two  functions,  V„_,  and  Vn+l,  have  then  for  x=b  values  different 
from  zero  :  moreover,  these  values  are  of  contrary  signs,  because  the  same 
equation, 

Va_1=VnQn-Vn+1, 

gives  V„_i  =  — Vn+n  when  we  have  Vn=0. 

*  This  depends  upon  a  principle  demonstrated  at  Art.  239,  Cor.,  that  if  a  function  of  a  be 
arranged  according  to  the  ascending  powers  of  v,  u  may  be  taken  so  small  that  the  sign 
of  the  whole  function  shall  depend  upon  that  of  its  first  term. 


326  ILttKBRA. 

This  being  established,  substitute  in  place  of     two  numbers,  b—u  and  / 
very  little  different  From  b;  the  two  functions,  S      ,  nod  Vn+i.  will  have  ton 
these  two  values  of  x  the  same  Bigns  as  they  have  for  x=zb,  sine  an  al- 

ways take  a  sufficiently  small,  to  insure  thai  neither  V„  ;  nor  V ,,+1  shall  chi 
sign  when  r  enlarges  in  the  interval  from  b —  u  to  A-}-//.     Whatevel  may  he 
the  sign  ot  V„  for  xssi —  u,  as  it  is  placed  in  the  iion  of  Bigns  between 

those  of  Vn_[  and  Vrn+1,  which  are  contrary,  the  Bigns  of  these  three  consecu- 
tive functions,  V„  ,.  Y„.  V  ,+!,  for  /='*  —  u,  will  form  always  either  a  perma- 
nence followed  hy  a  variation,  or  a  variation  followed  by  a  permanence,  as  i* 
seen  in  the  following  scheme 

V^  V„  Vn+1  V    ,  v„  vn+l 

For  X=b — U        -f-       rb       —,  or  else,      —      i      +• 

Similarly,  die  signs  of  the  three  functions,  Vn_i,  V„,  V,l+|,  for  .r  =  b-{-v, 
whatever  may  be  that  of  V„,  will  form  one  variation,  and  will  form  but  one. 

Besides,  each  of  the  other  functions  will  have  the  same  sign  for  X=b  — 
and  .r  =  l)-\-u,  provided  no  one  of  them   is  found  to  be  zero  for  x  =  b  at  the 
same  time  as  Vn. 

Consequently,  the  succession  of  the  signs  of  all  the  functions,  V,  Vj  ...  Vr, 
for  X=b-\-U,  will  contain  precisely  BS  many  variations  as  the  succession  of 
their  signs  for  x=b  —  U.  Thus,  the  number  of  variations  in  the  succession  of 
signs  is  not  changed  when  any  intermediate  function  whatever  passes  through 
zero. 

Oue  arrives  evidently  at  the  same  conclusion,  if  many  intermediate  functions, 
not  consecutive,  vanish  for  the  same  value  of  X.  But  if  this  value  should  de- 
stroy also  the  first  function,  V,  the  change  of  sign  of  this  one  would  then  make 
one  variation  disappear  at  the  left  of  the  succession  of  signs,  as  has  been  shown 
in  IV. 

VI.  It  is  then  demonstrated  that  each  time  that  the  variable  x,  in  increasing 
by  insensible  degrees,  attains  and  passes  a  value  which  renders  V  equal  to 
zero,  the  series  of  the  signs  of  the  functions  V,  V,,  Va  ...  V,  loses  a  varia 
tion  formed  on  its  left  by  the  siy;ns  of  V  and  Vt,  which  is  replaced  by  a  per- 
manence,  while  the  changes  of  signs  of  the  intermediate  functions.  V,,  V 
—  Vr_,,  can  never  either  augment  or  diminish  the  number  of  variations  which 
existed  already.  Consequently,  if  we  take  any  number  whatever.  A,  positive 
Or  negative,  and  any  other  number  whatever.  B,  greater  than  A,  and  if  we 
make  X  increase  from  A  to  P>.  as  many  values  of  X  as  are  comprised  between  A 
and  B,  which  render  V  equal  to  zero,  so  many  variations  will  the  succession 
of  signs  of  the  functions  V,  V,  ...  Vr  for  .''=1!  contain  less  than  the  suc- 
cession of  their  Bigns  for  ..'=  A.      This  Was  the  theorem  to  be  demonstrated. 

Remark. — In  the  successive  divisions  which  Berve  to  form  the  functions  V  . 
V3,  &c,  we  can,  before  taking  a  polynomial  for  a  dividend  or  divisor,  multiply 

or  divide  it  by  any  positive  number  at  pleasure.  The  functions  V,  V,,  V  . 
....  Vrl  obtained  by  this  operation,  will  differ  only  by  positive  numerical  fac- 
tors from  those  which  we  have  previously  considered,  and  which  appear  in 
equations  (l),  so  that  they  will  have  respectively  the  same  Bigns  as  these  for 

each  value  of  X.  , 

With  this  modification  we  can.  when  the  coefficients  of  the  equation  V=fi 
are  whole  numbers,  form  polynomials  V  .  V   .  &c.,  the  coefficients  of  w 
shall  be  also  entire,     lint  it  is  necessary  to  take  good  care  thai  the  num 
factors  thus  introduced  oi  suppressed  be  all  positive. 


THEOREM  OF  STURM.  327 

VII.  This  theorem  gives  the  means  of  knowing  the  whole  number  of  real 
roots  of  the  equation  V=0. 

In  fact,  an  entire  polynomial  function  of  x  being  given,  we  can  always  as- 
sign to  x  such  a  positive  value  as  that  for  this  and  eveiy  greater  value  the 
polynomial  will  have  constantly  the  sign  of  its  first  term  (see  Art.  239).  It  is 
the  same  with  all  negative  values  of  x  below  a  certain  limit.  All  the  real  roots 
of  the  equation  V  =  0  being  comprised  between  —  co  and  -\-<x>,  it  will  be  suffi- 
cient, in  order  to  know  their  number,  to  substitute  — co  and  -|-co  instead  of  A 
and  B,  iu  the  functions  V,  V,,  V2...  Vr,  and  to  note'  the  two  successions  of 
signs  for  —  co  and  -j-oo.  When  we  make  x=  +  co,  each  function  is  of  the 
same  sign  as  its  first  term.  For  x=  —  co,  each  function  of  an  even  degree,  in- 
cluding Vr,  has  the  same  sign  that  it  has  for  .r  =  -f-  co  ;  but  each  function  of  an  un- 
even degree  takes  for  x=  —  co  a  contrary  sign  to  that  which  it  has  for  x  =  +  co 
The  excess  of  the  number  of  variations  formed  by  the  signs  of  the  functions  V, 
V  x  ...  Vr,  for  x=  —  co,  over  the  number  of  variations  for  .r=-|-co,  will  express 
the  whole  number  of  real  roots  of  the  equation  V=0.* 

To  determine  the  initial  figures  of  the  roots,  we  may  substitute  the  sue 
cessive  numbers  of  the  series 

0,  —1,  —2,  —3,  —4, 

till  we  have  as  many  variations  as  — co  produced;  and  if  we  substitute  the 
numbers  of  the  series 


*  One  might  be  curious  to  know  how  the  succession  of  signs  of  the  functions  V,  VIf  V2 

.  .Vr  must  undergo  change  so  as  that  a  variation  is  lost  every  time  that  V  vanishes. 

We  have  seen  (IV.)  that  if  c  is  a  root  of  the  equation  V=0,  the  two  functions  V  and 
Vi  must  have  contrary  signs  for  .r=c — ?/,  and  the  same  sign  for  x=c-\-u.  So  that  if  we 
designate  by  d  the  root  of  the  equation  V=0,  which  is  next  greater  than  c,  so  that  be- 
tween c  and  d  there  is  no  other  root,  V,  will  have  for  x=zd — u  a  sign  contrary  to  that  of 
V.  But  V  has  constantly  the  same  sign  for  all  values  of  x  comprised  between  c  and  d  ; 
and  as  V!  has  the  same  sign  as  V  for  x=c-\-v,  and  a  contrary  sign  to  that  of  V  for  x=d 
— u,  we  see  that  Vj  has  two  values  with  contrary  signs  for  x=c-\-u  and  for  x=d — u  ; 
then,  while  x  increases  from  c-\-u  to  d — u,  YY  must  change  sign  once,  or  an  uneven  num- 
ber of  times  (I.,  or  Prop,  of  Ait.  252,  Cor.  1). 

Let  y  he  the  only  value  of  x,  or  the  least  value  of  x  between  c  and  d,  for  which  V, 
changes  sign.  V  and  V2  will  have  for  .T=y — u  the  same  common  sign  that  they  have  for 
a;=c-f-M.  For  a?=}'-f-w  V  will  have  this  same  sign  ;  but  V[  will  have  the  contrary  sign. 
V,  will  have  a  sign  contrary  to  that  of  V  for  the  three  values  for  y — v,  y,  and  y-f-w  (V.).  If, 
for  example,  V  is  positive  for  x=c-\-u,  we  have  the  following  table  : 

v  V,  vs 

For  x=y — u    +  -\ 

x—y          +   0  — 
x=y+u    -j 

Thus,  before  x  attained  the  value  c,  which  destroys  V,  the  signs  of  V  and  V!  formed  a 
variation  which  is  changed  into  a  permanence  after  x  has  overpassed  this  value  c ;  this 
permanence  subsists  until  Vi  changes  sign,  then  it  is  anew  replaced  by  a  variation  after 
the  change  of  sign  of  Vi ;  but,  at  the  same  time,  there  is  a  variation  fonned  by  the 
of  V]  and  Vj  which  changes  into  a  permanence,  so  that  the  number  of  variations  in  the 
total  succession  of  signs  is  neither  increased  nor  diminished. 

If  Vi  changes  sign  a  second  time  for  a  new  value  of  x  comprehended  between  c  and  d , 
the  variation  which  the  signs  of  V  and  V  form  before  x  attains  this  value  will  he  b 
replaced  by  a  permanence ;  and  still,  on  account  of  V;,  the  number  of  variations  will  re- 
main the  same  in  the  succession  of  signs.     As  Vi  can  thus  change  sign  only  an  uneven 
number  of  times,  after  its  last  change  the  signs  of  V  and  Vi  will  form  a  variation  which 

will    in itil      attains  the  valne  d,  which  destroys  V.    We  have  not  to  consider  here 

u,  e  V  va..i    -  is  without  changing  sign. 


328  ALGEBRA 

0,  1,  2,  3,  4,  

till  wo  arrive  at  a  cumber  which  pro  luces  as  vaaaj  variations  as  -|-ao  then 
the  numbers  thus  obtained  will  be  die  limits  of  the  roots  of  the  equation,  and 
the  situation  of  the  roots  will  be  indicated  by  the  signs  arising  fron  rhe  sub- 
stitution of  the  intermediate  numbers. 

We  shall  now  apply  the  theorem  to  a  few 

EXAMPLES. 

(1)  Find  the  number  and  situation  of  the  roots  of  the  equation 

x3— 4a:2— Gx+8=0.* 
Here  we  have  V  =  a? — 4a? — Gx-f-8 

V1=3.r2  — 8.r— 6; 
then,  multiplying  the  polynomial  V  by  3,  in  order  to  avoid  fractions, 
3x2_8z— 6)  3.1^— 12x2  —  18x+24  {x  —  1 
3.r»_   8.r-—   6x 

—  4x2— 12x-|-24,  multiply  by  J; 
or  —  3.C-—   9x-f  18 

—  3.r2+  8.r-r-   6 

—  17x+12  .-.  V2=17x— 12 
3j2_  8r_G 

17 


17x— 12)51x-— 13Gz— 102  (3x 
51x2— •   36* 

—  lOOx— 102. 
It  is  now  unnecessary  to  continue  the  division  further,  since  it  is  very  ob- 
vious that  the  sign  of  the  remainder,  which  is  independent  of  x,  is  —  ;  and, 
therefore,  the  series  of  functions  are 

V  =     Is—  4x"  —  Gx+8 
V1=  31s—  8x  —6 
VB=17z— 12 

V:;  =  +  - 

Put  -\-  co  and  — oo  for  x  in  the  leading  terms  of  these  functions,  and  tha 
signs  of  the  results  are 

*  The  process  applied  to  the  general  cubic  equation  a^-f-tf-r-'-f/u-f-c^O,  gives  the  fol- 
lowing functions,  viz. : 


With  the  second  A 

V  =  .??+  ax°~\-bx-\-c 

V,=3^-i-2r7j:  +b 

Va=2(a2— 3%-}-rt£— 9c | 

V3=— 4a3c+a2i=— \8abc— ilr>— 


0) 


Without  ;  m,  or  a—0. 

V  =  tf+bx+c 

V,=3.r*-f/> 

Y.i=—^bx—2c 

Va= — 16!— -J7r- 


(2) 


These  functions  in  (1)  and  |  J)  «  ill  frequently  he  {bond  useful  in  the  application  of  Sturm's 
theorem  to  equations  of  the  third  de         ,  i  the  ions  in  any  particular  ex- 

ample may  be  found  by  substitution  only.  In  order  that  all  the  roots  of  the  equation 
o?-\-bx-\-c=0  may  be  real)  the  first  terms  ei'  the  {auctions  must  he  positive ;t  hence  — 26* 
and  — 463 — 27c2  must  be  positive  :  and  as  — 27ca  is  always  negative,  b  must  he  negative, 
in  order  that — 4i3  and — 2b  may  he  positive;  therefore,  when  all  the  r.*>t-;  are  real,  4b1 

C'V  I   V 

must  be  greater  than  27c",  or  l-l    greater  than  I  ( )  •     V\  hen,  therefore,  £  is  negative  anil 


(')Hr 


3' 
all  the  roots  are  real,  a  criterion  which  has  been  long  known,  and  as  simple  as 


-..in  be  given. 

T   Si 


THEOREM  OF  STUIlil. 


329 


For      2-=-j-oo,  +  +  +  +  no  variation, 

■v=  —  », 1 1-  three  variations, 

.-.  3  —  0  =  3,  the  number  of  real  rbots  in  the  proposed  cubic  equatun. 

Next,  to  find  the  situation  of  the  roots  we  must  employ  narrower  limits 

than  -(-co  and  -co.     Commencing  at  zero,  let  us  extend  the  limits  both  ways. 

WiVjVs    Var.  VViViV.,    Var. 

2 


For.r  =  0  signs  -\ 1- 

2 

For  x=      0  s'iejds  -| \- 

.r=l....  --  +  + 

1 

x=-l....  +  +  -  + 

2=2.      .. +  + 

1 

x=-2....  -+-  + 

,r=3..     . +  + 

] 

x=4  .  .     . h  +  + 

1 

x=5....  +  +  +  + 

0 

We  perceive,  then,  by  the  columns  of  variations,  that  the  roots  are  between 
0  and  1,  4  and  5,  — 1  and  — 2  ;  hence  the  initial  figures  of  the  roots  are  — 1, 
0,  and  4  ;  and,  in  order  to  narrow  still  further  the  limits  of  the  root  between 
0  and  1,  wo  shall  resume  the  substitutions  for  x  in  the  series  of  functions  as 
before.  But  as  the  substitution  of  1  for  x,  in  the  function  V,  gives  a  value 
nearly  zero,  wo  shall  commence  with  1,  and  descend  in  the  scale  of  tenths, 
until  we  arrive  at  the  first  decimal  figure  of  the  root. 

Let  x=  1  signs 1-  -f-  one  variation, 

.r=-9  .  .  .  .  -| }-  -4-  two  variations  ; 

hence  the  initial  figures  are  — 1,  -9,  and  4. 

(2)  Find  the  number  and  situation  of  the  real  roots  of  the  equation 

^_|_.r3— x2— 2.r+4  =  0. 
Here  the  several  functions  are 

V  =       r»+  a?—  x"— 2:c+4 
V,=      4xs-\-3x°~—2x  —2 
V.2=        x*+2x  —6 
V3=-  x  +1 
V,  =  +  . 

Let     x=  -f-  qd,  signs  of  leading  terms  +  +  H H  lrsvo  variations 

.r=  —  co -| f-  -{-  -|-  two  variations  ; 

and  all  the  roots  of  the  equation  are  imaginary. 

When,  in  seeking  for  the  greatest  common  divisor  of  V  and  V\,  we  arrive 
at  a  polynomial  Vn  (for  example,  at  that  of  the  second  degree),  which,  put 
equal  to  zero,  will  only  give  imaginary  values  of  x,  it  is  not  necessary  to  cany 
the  divisions  further,  because  this  polynomial  Vn  will  be  constantly  of  the  snma 
sign  as  its  first  term  for  all  real  values  of  x  ;  for  if  it  gave  a  plus  sign  for  one 
value,  and  a  minus  for  another,  there  must  be  a  real  root  between.* 

(3)  Required  the  number  and  situation  of  the  real  roots  of  the  equation 

2x-'— llx2+8.r— 16=0. 
The  first  three  functions  are 

V=   2x4  —  ll.r2+8.r— 16 
V,=   4.i- —  llx+4 
V.  =  ll.r2  — 12x  +32; 

*  This  consideration  is  :»f  importance,  as  the  calculations  for  letemiining  the  functions 
v*.;,  V;j  are  long,  especially  toward  the  last,  on  account  of  the  magnitude  of  their  numerical 
coefficients. 


33C  ALGEBRA. 

and  the  roots  of  the  quadratic  1118— 12x+32=0  are  imaginary,  for  11x32 
X4  ia  greater  than  12s;  hence  V  must  preserve  the  Bame  sign  f  r  ever) 
value  of  .r,  and  the  subsequent  i'uixt i<ms  can  not  change  the  Dumber  of  varia- 
tions, for  a  variation  is  only  losl  by  the  change  of  the  sign  of  V.     Honce, 

For  rr=-f  assigns  +  +  +  ""  variation, 

,r  = —  x  .   .   .  -| \-  two  variations; 

and  the  proposed  equation  has  two  real  mots,  the  one  positive  and  the  othei 
negative,  since  the  last  term  is  negative.     (Prop.  VIII.,  Cor.  5,  p.  014.) 

When             ./■  =  ()  signs  —  +  +             x=      0  sigus  —  + -f- 
1=1.... r-  xsr— 1 h  + 


o 


-+  +  *=-2.... + 


i=3 1-  +  +  x=  — 3 ^ H. 

Hence  the  initial  figures  of  the  real  roots  are  2  and  —0. 

When  two  roots  are  nearly  equal  to  each  oOier. 
y\)  Find  the  roots  of  the  equation 

a-s+llz2  —  102.r-f  181=0. 
The  functions  are 

V=       r»+ll.r!  —  102x+181 
V,=     3x-+22.r  — 102 
V.  =  122.r  —393 
V3=+; 
and  the  signs  of  the  leading  terms  are  all  +  ;  hence  the  substitution  of  —  >- 
and  +ao  must  give  three  real  roots. 

To  discover  the  situation  of  the  roots,  we  make  the  substitutions 

.r  =  0  which  gives  -| \-  two  variations, 

a=l -\ 1- 

•r=2 + •  + 

ar=3 -4. 1- two  variations, 

x=l -| — J — | — j-  no  variation  ; 

he  ce  the  two  positive  roots  are  between  3  and  4,  and  we  must,  therefore, 
form  the  several  functions  into  others,  in  which  X  shall  .)0  diminished  by 
?,.     This  is  effected  by  Art.  251,  p.  315  ;  and  wo  get 

V  =       7/r,4-20y-— 9y-f  1 

V'x=     rw/-  +  -l(i_y  — 9 

V'„  =  122>/  —27 

v:i=+: 

Make  the  following  substitutions  in  these  functions,  viz. : 

y=  0  signs  -| \-  two  variations, 

y  =  -l  ...+__  + 

■?/=-2  .  .  .  -1 \-  two  variations, 

?/=-3  .  .  •  +  +  +  +  »"  variation  : 
bence  the  two  positive  roots  arc  between  3*2  and  ■"■■"•.  und  we  must,  again. 
transform  the  Inst  functions  into  others,  in  which  y  shall  bo  diminished  by  '- 
Rffecting  this  transformation,  we  have 

V"  =        2  -1-J0C.:  —  — :  -4--003 
V",=     V.: '+  11 -J:  —-88 
\  —    ;'■<■ 

V"3=  +  . 


THEOREM  OF  STURM.  33 

Let  z=  Q    then  signs  are  -| (-  two  variations, 

;  =  .01 -\ (- two  variations, 

z  =  -Q2 [-one  variation, 

z  =  -03 -p- -J*  -(-  +  no  variation  ; 

hence  we  have  3-21  and  3-22  for  the  positive  roots,  and  the  sum  of  tht  roots 
is  —11  ;  therefore,  —11 —3-21  — 3-22=  — 17-4  is  the  negative  root 

When  the  equation  has  equal  roots. 

255.  When  the  equation  has  equal  roots,  one  of  the  divisors  will  divide  the 
preceding  without  a  remainder,  and  the  process  will  thus  terminate  without  a 
remainder,  independent  of  x.  In  this  case,  the  last  divisor  is  a  common  meas- 
ure of  V  and  V,;  and  it  has  been  shown  (Art.  253,  Scholium,  p.  321)  that  if 
(a: — a{){x  —  a.,)"  be  the  greatest  common  measure  of  V  and  Vi,  then  V  is  di- 
visible by  (x —  «i)-(r  —  a2)8,  and  the  depressed  equation  furnishes  the  distinct 
and  separate  roots  of  the  equation,  for  Sturm's  theorem  takes  no  notice  of 
the  repetition  of  a  root.  The  several  functions  may  be  divided  by  the  great- 
est common  measure  so  found,  and  tthe  depressed  functions  employed  for  the 
determination  of  the  distinct  roots  ;  but  it  is  obvious  that  the  original  functions 
will  furnish  the  separate  roots  just  as  well  as  the  depressed  ones,  for  the  for- 
mer differ  only  from  the  latter  in  being  multiplied  by  a  common  factor  (29) ;  and 
whether  the  sign  of  this  factor  bo  -|-  or  — >  the  number  of  variations  of 
must  obviously  remain  unchanged,  since  multiplying  or  dividing  by  a  positive 
quantity  does  not  affect  the  signs  of  the  functions  ;  and  if  the  factor  or  divisoi 
be  negative,  all  the  signs  of  the  functions  will  be  changed,  and  the  number  of 
variations  of  sign  will  remain  precisely  as  before. 

Find  the  number  and  situati  in  of  the  real  roots  of  the  equation 
a" — 7.r'  -4- 1  C.r1  -4-  .r-  —  lG.r+4  =  0. 

By  the  usual  process,  wo  find 

V  =     x5  —  7.r'  -4- 1 3s3  +  x-  —  1 6.r + 4 

V,=  5.1-*— 28.r!+39x2-4-2.r  — 16 

Vn  =  ll.r"— 48.1'-— 51.r  +2 

V3=  3a:2—  8a: +4 

V4=     x—  2 

V6=0. 

Hence  x — 2  is  a  common  measure  of  V  and  V\ ;  and  if 

.r=  —  co  the  signs  are 1 1 four  variations, 

.r=  —  2 1 1 four  variations, 

•>'=-! 0+-  +  - 

X—      0 -1 (-  -\ three  variations, 

x=      1 \-  -| two  variations, 

x=      2 0  0    0  0   0 

x=      3 |--f"~f"  one  variation, 

x=      4 -j — J — [ — j — f-  no  variation. 

Therefore  we  infer  that  there  are  four  distinct  and  separate  roots  ;  one  is  — 1, 
for  V  vanishes  for  this  value  of  x ;  another  between  0  and  1 ;  a  third  is  2,  and 
a  fourth  is  between  3  and  4.  The  common  measure  .r — 2  indicates  that  the 
polynomial  V  is  divisible  by  (.r — 2)2 ;  and  hence  there  are  two  roots  equal  to 
2  (Art.  2"53,  Cor.  1) 


M2  ALGEBRA- 

ll  may  happen  that  one  of  the  functions,  V,,  Va  ...  Vr_i,  should  be  found 
zero  either  for  X=A  or  u  =  B.  In  this  case  it  is  sufficient  to  count  the  varia- 
tions which  are  found  in  the  succession  of  signs  of  the  functions  V,  V,,  V3 
...  Vr,  omitting  tlie  function  which  is  zero.  This  results  from  the  demonstra- 
tion in  Ail.  254,  V,  for  the  case  where  an  intermediate  function  vanish)    , 

When  the  number  of  the  auxiliary  functions,  V  ,.  \ '_..  ecc,  is  equal  to  the 
ee  of  the  equation,  as  is  ordinarily  the  ca  b,  in  consequence  <>f  each  re- 
mainder in  Booking  for  the  common  divisor  being  one  degree  less  than  the  pre 
ceding,  the  number  of  imaginary  roots  in  the  equation  may  be  found  by  the  fol- 
lowing rule  :  Tlie  equation  V  =  0  will  have  as  many  pairs  of  im  ■  roots 
as  there  are  variations  of  sign  in  the  succession  of  the  signs  of  the  first  terms  of 
the  functions  V,,  V2,  &c,  to  the  sign  of  the  constant  Vm  ita 

This  follows  from  tlie  fact  that  two  eon.  ecutive  functions,  V„_i,  Y„.  are 
the  one  of  an  even,  the  other  of  an  odd  degree.  Then,  if  the  two  timet  ions 
have  the  same  sign  for  :r  =  -|-cc,  they  will  have  contrary  for  .r=  —  c,  and 

versa.     So  thai  if  we  write  the  succession  of  signs  of  V,  V,,  V, V   .  for 

r  = —  co  and  for  x=-j-cc,  each  variation  in  the  one  succession  will  correspond 
to  a  pennant  nee  in  the  other.  Tims,  the  number  of  permanences  for  .>•=  — co 
is  equal  to  tlie  number  of  variations  for  .r=-r-cc. 

But  for  .r=-f-co  the  number  of  variations  will  be  that  of  the  first  terms  of 
the  functions  V,  V,  . . .  V„„  which  denote  by  i.  Then  there  will  be  i  per- 
manences for  .r=  —  co  and  m  —  i  variations.  The  excess  of  the  number  of 
variations  m — i  for  x=  —  co  over  the  number  i  for  x=-f-co,  is  m — 2t,  which 
is  therefore  tlie  number  of  real  roots  of  the  equation,  and  therefore  -J/  the 
number  of  imaginary  roots,  the  whole  number  of  roots  being  m. 

horner's  method  of  resolving  numerical  equations  or  all  okokrs. 

256;   The  method  of  approximating  to  the  roots  of  numerical  equations  of 
all  orders,  discovered  by  W.  G.  Horner,  Esq.,  of  Bath,  England,  is  a  pi 
of  very  remarkable  simplicity  and  elegance,  consisting  simply  in  a  Bucce 
of  transformations  of  one  equation  to  another,  each  transformed  equation 

9  having  its  roots  less  or  greater  than  those  of  the  preceding  by  the  cor- 
responding figure  in  tlie  root  of  the  proposed  equation.  We  have  shown  bow 
to  discover  the  initial  figures  of  the  roots  by  the  theorem  of  Stobm  :  and  by 
making  tho  penultimate  coefficient  in  each  transformation  available  as  a  trial 
or  of  the  absolute  term,  we  arc  enabled  to  discover  the  succeeding  figure 
of  the  root;  and  thus  proceeding  from  one  transformation  to  another,  we 
enabled  to  evolve,  one  by  one,  the  figures  of  the  mot  of  the  given  equation, 
and  push  it  to  any  degree  of  accuracy  required. 

GENERAL   RL'I.l   I, 

1.  Find  tho  number  and  situation  of  the  roots  by  Snirm's  theorem,  and  let 
the  root  required  to  be  found  l>e  positive. 
•_'.  Transform  the  equation  into  another  whose  roots  shall  be  less  than  those 

of  the  proposed  equation  by  the  initial  figure  of  the  root. 

.;.   I  divide  the  absolute  term  of  the  transformed  equation  by  the  trial  divisor, 

or  penultimate  coefficient,  and  the  n.  \t  figure  of  the  root  will  be  obtained,  by 
which  diminish  the  root  of  the  transformed  equation  as  before,  and  proceed  in 
this  manner  till  tho  root  bo  found  to  the  required  accUTI 


NUMERICAL  SOLUTION  OF  ALGEBRAIC  EQUATIONS.  333 

Note  1. — When  a  negativo  root  is  to  be  found,  change  the  signs  of  the  alter- 
nate terms  of  the  equation,  and  proceed  as  for  a  positive  root. 

Note  2. — When  three  or  four  decimal  places  in  the  root  are  obtained,  the 
operation  may  be  contracted,  and  much  labor  saved,  as  will  be  seen  in  tho 
following  examples : 

EXAMPLES. 

(1)  Find  all  the  roots  of  the  cubic  equation 

3?—1tx-\-  7=0. 
By  Sturm's  theorem,  the  several  functions  are  (Note,  p.  328), 

V  =  a*— 7z+7 

V,  =  3x~  —  7 
Vi=2x  —3 

Hence,  for  x=-\-fx>  the  signs  are  +  +  +  +  no  variation, 

x=  —  oo 1 \-  three  variations  ; 

therefore  the  equation  has  three  real  roots. 

To  determine  the  initial  figures  of  these  roots,  we  have 

for  x=0  signs  -| (-  for  :r  =      0  signs  -| 1- 

x=l  .  .  .  -\ 1-        *=— 1  •  •  •  -\ h 

x=2...+  +  +  +        x=-2  .  .  .  +  +  -  + 

x=-3  .  .  .  +  +  -  + 
x=-4  .  .  .  -  +  -  + 
nence  there  are  two  roots  between  1  and  2,  and  one  between  — 3  and  — 4 

But  in  order  to  ascertain  the  first  figures  in  the  decimal  parts  of  the  two 
roots  situated  between  1  and  2,  we  shall  transform  tho  preceding  functions  into 
others,  in  which  the  value  of  a:  is  diminished  by  unity.  Thus,  for  the  fui  ction 
V  we  have  this  operation : 

1_|_0   —7   +7  (1 
1        1   —6 
1  ^6  +T 

1  2 

2  ^4 
1 

~3 
And  transforming  the  others  in  the  same  way,  we  obtain  the  functions 
V  =f+3y*— 42/+1;  V\=3y*+6y— 4;  V'3=2y— 1;  V'.«  =  +  . 

Let  2/=*1  t^ien  tf10  SI8ns  nre  ~\ 1~  two  variations, 

y  =  -2 -\ 1-  do. 

y=-o + +  do. 

y= -4 1-  one  variation, 

2/  =  -5 T+  do- 

2/-6 -  +  +  +  do. 

y  =  -7 +  +  +  "f*  no  variation. 

Therefore,  tho  initial  figures  of  the  three  roots  are  1-3,  1-6,  and  — 3. 

The  rest  of  the  process,  with  a  repetition  of  the  above,  is  exhibited  and 
afterward  explained  below. 


334 


ALGEBRA 

1  +  0 

1 

1 
1 

—  7 
1 

—  6 
o 

+  7  (1  -35689580  7 
—6 

•1... 

—903 

o 

1 

—  *4.. 
99 

*97  .  .  . 

—  86625 

•33 
3 

—  301 
108 

—  *1  9  3 

1975 

•10375  . . 

—9048984 

36 
3 

•1326016 

—  1181430 

*39  5 
5 

—   17325 
2000 

36 

141586 
—  132923 

40  0 
5 

—•15  3  25 

243 

8663 
—7382 

*40  56 
6 

—   1508164 
24372 

1281 

—  1181 

40  62 

6 

—  *1  48379 
3  25 

g 

4 

8 
4 

100 

—89 

*j40|68 

—   148053 
325 

11 
—  10 

—    147  7218 
3|6 

1 

—    1  4  7  G  9 
3 

•> 
6 

—  1|4|7|6|5 
The  process  here  is  similar  to  that  on  p.  318.  Tht<  numbers  marked  with 
stars  are  the  coefficients  of  the  equation  having  the  reduced  roots.  Thus,  *3, 
*4,  and  *1  are  the  coefficients  of  the  equation  whose  roots  are  1  less  than 
those  of  the  proposed  equation.  The  right-hand  3  of  *3C  he  .">  tenths  add- 
ed in  the  next  step  of  the  process,  which  has  for  its  object  to  reduce  the  roots 
by  -3.  The  coefficients  of  the  resulting  equation  are  *39,  — *193,  and  *97. 
Now,  instead  of  going  on  in  this  manner  to  obtain  the  following  figures,  568, 
dec,  of  the  root,  lite  method  of  proceeding  changes  :  the  193,  which  is  t lie 
penultimate  coefficient,  becomes  a  trial  divisor,  by  which  dividing  the  absolute 
term  97,  which  is  .097,  the  divisor  being  1-93,  the  quotient  is  5,  the  next  fig- 
ure of  the  root,  which  is  .05.  This  5  is  annexed  to  the  '•".!>.  and  we  proceed 
as  before;  that  is,  multiply  the  *395  in  the  first  column  by  this  5,  producing 
•1975 in  the  second  column,  and  by  addition,  1*7325,  and  so  on.  To  show  that 
the  quotient  figure  5  is  obtained  by  means  of  the  trial  divisor,  observe  that  the 
1-7325  is  nearly  equal  to  the  U-:i3  above,  and  that  the  -086625  in  the  third 
column,  which  is  the  product  of  1*7325  by  the  -05,  is  nearly  equal  to  tile  *-097 
above;  hence  the  quotient  of  ••  097  by  1*93  is  nearly  this  same  '05. 

The  further  W6   DTOCeed,  the   inure    accurate  this   process  becomes,  for  the 

first  figure  of  each  Dumber  in  the  first  column  being  units,  this,  multiplied  by 
the  decimal  figure  found  in  the  root,  which  is  thousandths,  tens  of  thousandths, 

and  so  on,    that  is.  soon  a  very  small  fraction.  gives  thousandths,  tens  of  thou- 
sandths, and   so  on,  or  a   \eiy  small   fraction,  for  the    product  :   atnl.  the  first 


NUMERICAL  SOLUTION  OF  ALGEBRAIC  EQUATIONS. 


335 


liixire  in  the  numbers  of  the  second  column  being  also  units,  these  numbers 
tire  not  much  affected  by  tho  addition  of  the  above-named  products.* 

When  the  number  of  decimal  places  in  the  numbers  of  the  third  column 
hecomes  equal  to  tho  number  of  decimal  places  required  in  the  root,  il  will 
not  be  necessary  to  obtain  any  more  in  the  third  column  ;  and  as  each  new 
decimal  figure  in  the  root,  multiplied  by  the  number  in  the  second  column, 
would  make  one  more  place  in  the  third,  it  will  be  necessary  to  cut  off  one 
figure  in  the  second  column,  and,  for  a  similar  reason,  two  figures  in  the  first 
column.  As  soon  as  the  figures  are  all  cut  off  in  the  first  column,  the  process 
becomes  simply  one  of  division,  the  divisor  and  dividend  rapidly  diminishing. 

Wo  have  thus  found  one  root  x=l-356895867 ,  and  the  coefficients 

of  the  successive  transformed  equations  arc  indicated  by  the  asterisks  in  each 
column.     To  find  another,  we  have  the  following  : 

1  +  0  —7  +7(1-692021471 

1  1  -G 


1 
1 

2 

1 


-G 
2 

•4  .  . 
216 


36 
6 

42 
6 

48  9 
9 

49  8 
9 

50  72 
2 

50  74 
2 

1... 

■1104 

•   104... 
100809 


■  18  4 
252 


68  .  . 
44  01 


112  0  1 
4482 

15  6  8  3.. 

10  144 

1578444 
10148 

15885 

1 


-3191... 
3156888 

—34112 

31774 

—2338 
1589 

—749 
635 

—  114 
111 


|50|76  1|5|8|8|7| 

Another  root  is  .r=l-692021471   .  .  . 


For  the  negative  root,  change  the  signs  of  the  second  and  fourth  terms. 


*  To  show  tliis  in  a  more  general  way,  let 

aa;n+Bxn-I+B.zn-- ....  +Bn_,.r+Bn=0 

oe  one  of  the  depressed  equations  which  is  to  furnish  the  next  decimal  place  of  the  root  of 
,he  proposed  equation;  the  value  of  a;  in  this  depressed  equation  will  of  course  be  a  very 
jraftU  fraction;  hence  the  higher  powers  of  it  may,  without  much  error,  be  neglected.  The 
,1. •pressed  equation  thus  reduces  to 

B^.t+B^O. 

Hence  the  value  of  x,  without  regard  to  its  sign,  is 

x- 


"Bn 


nearly  ;  that  is,  it  may  be  obtained  by  dividing  the  ultimate  by  the  penultimate  coefficient 


336 


ALGEBRA. 


-0 
3 

3 

3 

~6 
3 


7 
9 


90  4 

4 

90  8 

4 

91  28 

8 

9136 

8 

.|91|44 


2 

18 

20  ...  . 
3616 

2,4 

203616 
3632 

2  0  7  2  4  8 
730 

20797824 
73088 

2087091 
823 

2 

0 

2087914 
823 

2 

0 

208873 

7 
9 

208874 

6 
9 

— ?  (3-048917339* 

+  6 

—  1 

814464 


—  185536... 
166382592 

—  19153408 
18791228 


—362180 

208875 

—  153305 

146212 


—  7093 
6266 

—  827 
626 

—  201 
188 


—  13 
L2 


2|0|8|8|715  1 

Hence  the  three  roots  of  the  proposed  cubic  equation  are 

x=      1-356895867 

x=      1-692021471 

x=—  3-048917339 

(2)   Find  the  roots  of  the  equation  .r3-f  ll.r3— 102x+ 181  =  0. 
We  ltave  already  found  the  roots  to  be  nearly  3-21,  3*22,  and  —17. 
1  b  4,  p.  330.) 


+11 

3 

—102 
42 

page. 

+  181  (3-2131277.0 
—  180 

14 
3 

—  60 
51 

1... 

—  992 

1  7 
3 

—     9  .  . 

4  04 

8... 
—  6739 

2  02 
o 

—     496 

408 

1261... 

—  1217403 

■jiil 
2 

—        88.  . 
2061 

43597 
--34183 

2  06  1 
1 

6739 
2  0  6  '.' 

9414 

—  <;787 

2  06  2 

—       4677  .  . 
Carried  to  next 

2627 

(See 


NUMERICAL  SOLUTION  OF  ALGEBRAIC  EQUATIONS. 


a37 


2  Oi 

2  06  33 
3 

2  06  36 
3 

4677  .  . 
61899 


•2106139 


—   405801 
61908 

—   343893 
2064 

—   341829 
206  4 

—   3397  6 
.  4  1 

—   3393 
4 

5 

1 

2627 
—2372 

255 
—  237 

18 
—16 


3|3|8|9 


In  a  similar  manner,  the  two  remaining  roots  will  be  found  to  be 


and 


.r=3-22952121 


x=  — 17-44264896. 


(3)  Given  a-4-|-:r',-{-t2+3:r — 100=0,  to  find  the  number  and  situation  of 
the  real  roots. 
Here  we  have 

V  r-  .r'+  x3-f-  a:2_j_3x_100 
V1^=4x5+3x^2x  -\-3 
V2=— 5.r2— 34a:+1603 
V3=—  1132.r+6059 
V,  =  — 

Iiet  a'=  —  co  then  signs  are  -| 1 three  variations, 

.r=-4-co +H one  variation  ; 

hence  two  roots  are  real  and  two  imaginary ;  and  the  real  roots  must  have 
contrary  signs,  for  the  last  term  of  the  equation  is  negative.     To  find  th  3  si^ 
nation  of  the  roots 

in  V  ViVsVsV, 

Let  x=0  signs \--\--\ 

x=l.  .  ._+++_ 
x=2.  .  ._+++- 
x=3.    .   .+  +  +  +  - 

in  V  V,V9V,V< 

Also,  x=      0  signs h4H 

.r=—  1  •    •   •  —  0  +  +  — 

x=-2.   .   . +  +  - 

T=_3.   .   . +  +  - 

x=-4.   •   .+-  +  +  _ 
In  this  example  the  function  Vx  vanishes  for  .r= —  1,  and  for  the  o^.... 
value  of  x  the  functions  V  and  V2  have  contrary  signs,  agreeably  to  V.,  p 
325,  and  writing  -4-  or  —  for  0  gives  the  same  number  of  variations.     The 
initial  figures  of  the  root  are,  therefore,  2  and  — 3. 

Y 


338 


ALGEBRA. 


To  find  the  negative  root,  we  have  the  following  operatiou 


11  4 
4 


11  8 
4 


12  2 

4 

~ n]~63 

3 

12  66 

3 

12  69 

3_ 

12  723 

3 

"  12  726 

3 

~~ 12  72!» 

3 

|-121732 


I— 1 

3 

2 

+  1 

6 

7 

3 
5 

15 
22 

3 

8 

24 
~46  .  . 

3 

4  56 

50  56 

4  72 

55  28 

-!  88 

60  16  .  . 

37  89 

60  53  89 

37  98 

60  91  87 

38  07 

61  29  91 

3  81  69 

61  33  75  69 

3  8178 

61  37  57  47 

3  81  87 

61  41  39 

34 

63 

6 

61  42  02 

9 

63 

6 

61  42  66 

5 

63 

6 

|61|43|30 


21 

1  8 

66 

84  .  .  . 

2  0224 

10  4  2  2  4 

2  2  112 

12  6  3  3 6 

1816167 

128152167 

18  2  7  5  6  1 

12  9  979728.   .  . 

184012707 

13016  3  740707 
184127241 


034786794 
3  0  7  10  14 


1 3037857809 
3071332 


13  0  4  0  9  2  9  14 
43003 


1304135  9  17 
4  3  0  0  3 


13  0417892 
430 


13  0  418322 

430 


13041875 
4 

■ 

1 

3  ii 

i 

1 

.- 

-  u 
I4 

—  100  (3-433577863365M 


—46.    .. 
416696 
—  43104... 
384456501 

390491222121 

~; 

651892890  16 

—  10154478833 

9128951421 

—  1025527 

91292 

—  112599158 

101335040 


14118 

.130 
— l: 
391 

-47732 
391 


— - 


—781 

—  129 

117 


—  12 
II 


1 


For  the  positive  root  wo  have  a  similar  operation, 

1  +1  +1  +3  —100  (2-80285121815- .': 
hut  this  wo  shall  leave  for  the  student  to  perform,  and  the  two  roots  will  b« 
found  to  be 

x=      2-8028512181582  .  .   . 

x=—  3-4335778633659  .   .  . 

(1)  Find  tho  roots  of  the  equation  a-5+2.( •'  +  ."..< "'+  1  r'-f">.r— 20=0 
Here  we  have  V  =  a*+  2r,-f-  3r>+  I r- +  5.r  —  20 
V,=5.r,4-  8a?4-  9z"+ar+5 
\    —  —  7./-' — 2 1  ./-•  —  1 2  x  +  255 
V3=  — 13r+14 
V4=  — 

For  r  = — j.   we  have  signs (-  +  H ,NV"  varialione  ; 

tzzz-^-cc -|"H um;  variation. 


NUMERICAL  SOLUTION  OF  ALGEBRAIC  EQUATIONS. 


3W 


Hence  the  difference  of  variations  of  sign  indicates  the  existence  of  one  real 
-ind  four  imaginary  roots,  the  real  root  being  situated  between  1  and  2. 


1  +  2 

4-  3 

+   4 

+  5 

—  20  (1-125790.. 

1 

3 
6 

6 

10 

15 

3 

1  0 

15 

1 

4 

1  0 
20 

20 
35  .  . 

387171 

4 

1  0 

—  112829 

1 

5 

15 

37171 

87005 

5 

15 

o  0  •  •  • 

38  7  1  7  1 

—  ■25824 

1 

6 
21.. 

2171 
3  7171 

3  9  4  14 

22285 

6 

42  6  5  8  5 

—3539 

1 

71 

22  43 
3  94  14 

844 

3136 

71 

2171 

43  5  0  2 

5 

—403 

1 

72 
2  243 

23  1G 
4  17130 

8534 

403 

72 

44  3  5 

ii 

1 

73 
2316 

4  7 

21 

•"> 

73 

4  22 

0 

44  5  7 

1 

1 

74 
|..2|390 

4 

7 

21 

5 

74 

4  26 

7 

44  7 

8 

1 

4 

7 

2 

I-  -75 


.4|31 


4418 


Hence  the  real  root  is  nearly  1-125790  ;  and  by  using  another  period  of  ciphers 
we  should  have  the  root  correct  to  ten  places  of  decimals,  with  very  little  ad- 
ditional labor. 


ADDITIONAL   EXAMPLES   FOR  PRACTICE. 

(1)  Find  all  the  roots  of  die  equation  r3 — 3.r— 1=0. 

(2)  Find  all  the  roots  of  the  equation  x1 — 22a: — 24=0. 

(3)  Find  the  roots  of  the  equation  a^-f-.r-— 500  =  0. 

(4)  Find  the  roots  of  the  equation  2?-\-a?-\-x — 100  =  0. 

(5)  Find  the  roots  of  the  equation  2r!-f-3.r2  — 4.r  — 10  =  0. 
(G)  Find  the  roots  of  the  equation  r'— 12.i--j-12.r— 3=0. 

(7)  Find  the  roots  of  the  equation  r1  —  8.r,-T-14.x'2+4.r— 8  =  0. 

(8)  Find  the  roots  of  the  equation  x* — x*-\-2x*-\-x — 4  =  0. 

(9)  Find  the  roots  of  the  equation  .r  —  10.r?4-6.r-r-l=0. 

(10)  Find  the  roots  of  the  equation  .r'4-3.r,4-2r''— 3x2  — 2.r  — 2  =  0 

(11)  Find  all  the  roots  of  the  equation 

a<U  4rr'  —  3x4  —  16.r"+llr--|-l  2.r— 9  =  0. 


ANSWERS. 


(x=  + 1-879385242 

(1)  \  x=— 1-532088886 
|^.r=  —  -347296355 

(x=  +  5-162277660166 

(2)  \  .r=  — 1-162277660166 
[,r=— 4 

(3)  a-=7-61727975593e 

(4)  x=4-264429973156 

(5)  x=l-6248190834°4 


(6) 


(7) 


f.r=  +  2-858083308178 
ar=4-  -606018306959 
x=+  -443276939592 
x=  —  3-907378554730 

(x=-f  5-236067977500 
J  x=+  -763932022500 
)  x=  -j-  2-732050807569 
[x=—  -732050807569 
$x=  +  l-146994592039 
).r=  — 1-090593586698 


:nO  ALGEBUA. 


ra:=— 3-06531579] 
x=  —   -691576280490080 

(9)    \  .(  =  —  -i: 

x=4-   •879508708414-li;n 
x=  +  3-0530581<- 


(10)  ssrl-0591090034618 

(  x=—l  ;  x=—  3;  .t  =  1 

(11)  <J  ar=— 3?  x=l 

x=l 


i 


257.  The  theorem  of  Sturm  gives  a  simple  means  of  establishing  the  cod 

mis  of  the  reality  nl'  the  roots.     As  the  real  roots  are  comprised  between 
two  limits,  — L'  and  +  L,  the  one  negative  and  th<-  other  positive,  which  n 
he  chosen  as  large  as  we  please,  the  question  reduce-  to  seeking  the  condition;' 
necessary,  in  order  that  from  x  = — L'  to  x=-{-h  the  series  V,  V  ,  Y  .  \c, 
should  lose  a  number  of  variations  equal  to  the  degree  of  the  equation. 

Supposing  this  degree  to  be  to,  it  must  then  lose  m  variations.  But  in  order 
that  it  may  have  m  variations,  it  is  necessary  that  it  should  have  at  least  rn-^-l 
1  as  it  can  not  have  more,  we  are  sure  that  the  quantities  V,  Y,,  V;, 
&c,  exist  to  the  number  m-j-1,  and  that  they  are  respectively  of  the  degree 
m,  m  —  1,  to  —  '2,  &c.  The  last,  which  does  not  contain  x,  will  then  be  repre- 
sented by  Vm. 

When  in  the  polynomial  functions  of  x  we  substitute  very  large  numbers, 
positive  or  negative,  for  X,  we  know  t lint  the  results  are  of  the  same  sign  as  if 
each  polynomial  were  reduced  to  its  first  term  :   therefore,  in  the  present  in 
vestigation,  we  need  occupy  ourselves  only  with  the  first  term.     Let  us  take 
the  equation  V  =  0  under  the  ordinary  form 

xm-\-pxm-1-\-qxm-*+,  &c.,  =0. 

The  first  term  of  Y  is  .rm;  that  of  the  derived  polynomial,  Y1(  will  be  mxm~ l. 
With  regard  to  those  of  tho  polynomials  V;,  Y3,  &c,  they  are  functions  com- 
posed of  the  coefficients^,  q,  &c,  determined  by  tho  successive  divisions  in 
accordance  with  the  rule.  Let  us  represent  these  functions  by  < ra,  <  ! . . .  .  Gm 
and  write  in  order  the  m-f-l  quantities, 

.rm,  mr™-1,  G.2xm-2,  G3xm~3  . . .  Gm. 
The  question  will  be  reduced  to  finding  the  conditions  which  will  cause  the 
loss  of  to  variations  from  this  series  when  we  pass  from  .)•=:  —  L'  to  .?•=-)- L. 
In  order  that  this  may  bo  the  case,  it  must  have  m  variations  upon  tho  substi- 
tution of  — L\  and  m  permanences  upon  the  substitution  of  +L.  But  in  this 
series  the  powers  of  .r  go  on  decreasing  by  unity  ;  consequently,  if  it  has  noth- 
ing but  permanences  when  .r  =  -j-L,  it  will  have  nothing  but  variations  when 
x=  —  L'.  Thus,  the  conditions  are  reduced  simply  to  such  as  require  this 
series  to  have  only  positive  coefficients,  that  is  to  say.  to  the  following, 

Gs>0,  G3>0  ....  Gm>0. 

These  conditions  will  never  be  greater  in  number  than  m  —  1,  but  they  may 
be  less  in  Dumber,  inasmuch  as  soino  of  the  above  inequalities  may  involve  the 
ere. 

KXAMI : 

.'".-.    Find  the  conditions  necessary  for  the  reality  of  the  roofs  of  tl 

.-  -\-,jr-\-r=Q. 
I  [ere  we  have  //<=:;,  and  the  conditions  an-  only  two  in  number,  G9>0  and 

<;,>o. 
To  find  <!:  and    : ;.  we  calculate  V.  ami  V    i  ■.  sive  divisions,  as  Pol. 

low 


RULE  OF  DES  CART:  341 

First  Division.  ^vision. 


•<.-'+   qx+   r 
■)i  -\-?,ijx-\-^r 


W+q  3a*+     q 


—   2qX—   -:.r 


1-j,   ■  4-    !,/•      _   6qx+   \)r 

3.r3+   qx  1-V  +1- 

2qx  +  3r  —  ltiqrx  + 

.  .  Vr,  =  —  2or—  3/-.  —  lfyrr  —  2 


.-.  V4=  —   4<y3  — 27r2. 
Consequently,  the  inequalities  G,.>0,  G3>0,  become 

_-.\7>0,  _4g3— 27r3>0; 
observing,  however,  that  the  fust  inequality  is  embraced  in  the  second,  since 
r*  is  always  positive;  and  changing  the  signs  of  the  second,  •we  have  for  the 
sole  condition  of  the  roots  of  an  equation  of  the  third  degree,  being  real, 

-h/'-f -27/--<0. 


We  have  now  given  so  much  of  the  general  properties  of  equations  of  all 
ees,  and  such  moles  of  proceeding,  as  will  insure  their  numerical  solution 
iu  a  manner  the  most  certain  and  infallible,  and  ordinarily  the  best. 

There  are,  however,  many  transformations  of  equations,  which,  by  i 
their  degree,  or  by  giving  them  a  particular  form,  serve  to  facilitate  their 
tion- in  certain  cases.     There  are  also  many  general  principles  applicable  to 
the  resolution  of  equations  of  the  higher  orders  by  the  methods  in  use  previ 
ous  to  the  discovery  of  Sturm,  which,  with  these  methods  themselves,  it  is  de- 
sirable to  know  for  many  purposes  in  the  application  of  algebraic  analysis  to 
the  higher  launches  of  both  pure  and  mixed  mathematics,  for  ulterior  improve-; 
menls  in  the  general  theory  of  equations  itself,  and  even  for  use  in  the  .- 
tion  of  equations,  in  some  cases,  to  which  they  are  more  conveniently  adapted 
than  the  method  of  Sturm.      A  treatise  on  algebra  could  scarcely  be  regard: 
as  complete  without  some  notice  of  these.     We  shall  therefore  give  as  extei 
sive  an  exhibition  of  them  as  can  in  any  way  be  useful  in  an  elementary 
like  the  present,  commencing  with  the  well  known 

RULE  OF  DES  CARTES. 
•2o*J.  An  equation  can  not  have  a  greater  number  of  positive  roots  than  there 
arc  variations  of  sign  in  the  successive  terms  from  -4-  to  — ,  or  from  —  to  -{-. 
nor  ran  it  have  a  greater  number  of  negative  roots  than  there  are  permanent,  • . 
or  successive  repetitions  of  the  same  sign  in  the  successive  U  rms. 

Let  an  equatiou  have  the  following  signs  in  the  successive  terms,  viz.-*: 

+  -+ +  +  +  ->or+ +  _  +  +  +  . 

Nov.    if  we  introduce  another  positive  root,  we  must  multiply  the  equation  by 
r — a,  and  the  signs  in  the  partial  and  final  products  will  be 

-H —  +  +  + +  -+++-+ 

4- — 1 — ±±  +  ±± — h  +-±±  +  -+±±- 

where  the  ambiguous  sign  Az  indicates  that  the  sign  may  be  -f-  or  —  accord 
»ng  to  the  relative  magnitudes  of  the  terms  with  contrary  signs  in  the  partial 
products,  and  where  it  will  be  observed  the  permanences  in  the  proposed 


3 12  AL<. 

equation  are  changed  into  signs  of  ambiguity;  hence  the  permanences,  tui^ 
the  ambiguous  sign  as  yon  will,  are  not  increased  in  the  final  product  by  Hie  in- 
troduction of  the  positive  root  -\-a  ;  but  the  number  of  signs  is  increi 
one,  and,  therefore,  the  Dumber  of  variations  must  be  increased  by  one.    He 
it  is  obvious  that  the  introduction  of  every  positive  root  also  introduces  • 
additional  variation  of  sign,  and,  therefore,  the  whole  number  of  positive  1 
can  not  exceed  the  number  of  variations  of  signs  in  the  successive  terms  of  the 
proposed  equation. 

Again,  by  changing  the  signs  of  the  alternate  terms,  the  roots  will  be  changed 
from  positive  to  negative,  and  vice  versa  (see  Prop.  VII.).  Moreover,  by  tin- 
change  the  permanences  in  the  proposed  equation  will  be  replaced  by  varia- 
tions in  the  changed  equation,  and  the  variations  in  the  former  by  perm 
in  the  latter;  and  since  the  changed  equation  can  not  have  a  greater  number 
of  positive  roots  than  there  are  variations  of  sign,  the  proposed  equation  can 
not  have  a  greater  number  of  negative  roots  than  there  are  permanences  ol 
sign. 

Let  v  be  the  number  of  variations,  c'  tin'  number  of  variations  of  the  trans- 
formed equation  obtained  by  changing  x  into  — ar.     The  number  of  real  ro 
of  the  equation  can  not  surpass  r-\-r'.     Then,  if  this  sum  is  less  than  the  de 
gree  m,  the  equation  will  have  imaginary  roots. 

The  sum  v-\-v'  is  never  gi  than  the  degree,  and  when  it  is  less  the 

difference  is  an  even  number.     (See  Art.  248.) 

EXAMINES. 

(1)  The  equation  .z-6+3.i-5  —  41r1  —  87r,+400.r2-(-44-U-— 720=0  has  six  real 
roots.     How  many  are  positive  ?  • 

(2)  The  equation  .r4— 3.v>  —  15.r--f  49.r  — 12  =  0  has  four  real  roots.  How 
many  of  these  are  negative  ? 

2G0.  We  give  next  the  repetition  of  a  principle  already  presented,  but  which 
may  be  derived  as  a  direct  consequence  of  the  theorem  of  Sturm. 

THEOREM  OP  ROLLE. 

Let  F(.r)  =  0  be  an  equation  which  has  no  equal  roots,  F '(.'')  its  derived 
polynomial.  We  have  seen  that  as  .r  increases,  the  series  of  Sturm  loses  a 
variation  every  time  that  x  passes  over  a  root  of  the  equation  F(.r)=0.  and 
that  it  can  not  lose  one  in  any  other  way.  Moreover,  we  have  seen  that  this 
variation  is  lost  at  the  commencement  of  the  series  of  functions,  in  conse- 
quence of  F(.r)  changing  sign,  while  F'(ar)  does  nut  ;  so  that  F(.r)  is  always 
of  a  sign  contrary  to  that  of  F'(ar)  for  a  value  of  x  a  little  less  than  the  root. 
and  always  of  the  same  sign  for  a  value  a  little  greater. 

Thus,  when  we  ascend  from  a  root  r  to  a  root  /■',  which  is  immediately 
above  r,  F(j  must  be  of  the  same  sign  as  F'(')  for  a  value  of  x  a  little  greater 
than  r,  and  of  a  sign  contrary  to  F '(.'")  for  a  value  of  a;  a  little  less  than  ;'.  Hut 
in  the  interval  F(.r)  docs  not  clue  >;  then  F'(ar)  must  change  sign  at 

least  once ;  therefore  the  equation  F'(x)=0  has  at  least  one  root  between  •• 
and  r'. 

I ,    I  a,  b,  c,  d  .  .  .  g  bo  the  real  idols  of  F(x)=0,  arranged  in  order  of  iim 

tude,  beginning  wid  the  largest ;  and  let  a,,  o,,  c,  .. .  g-,  be  the  real  roots  of 

K7  ;)  —  (>.  disposed  in   tho.-anie  manner.      We  have    jn-t   seen    that  tie 

are  comprised,  some  between  /;  and  h.  some  between  ''  and  c,  dec. ;  but  as  tha 


THEOREM  OF  110LLE  343 

degree  of  V'(x),  and,  consequently,  the  number  of  its  roots,  is  one  less  than 
the  degree  and  number  of  roots  of  F(.r)  =  0,  it  follows  that  the  equation 
F(.r)  =  0  can  have  but  one  root  above  a,,  but  one  between  a,  and  6,  .  . .,  and, 
finally,  but  one  below  gv  This  property,  which  has  been  long  known,  and  of 
which  we  have  given  an  independent  demonstration  at  (Art.  253),  is  identical 
With  the  theorem  of  Rolle. 

2G1.  The  considerations  which  lead  to  the  theorem  of  Itolle  furnish  also 
the  means  of  determining  whether  the  m  roots  of  the  equation  F(.r)=0  art- 
real  and  unequal. 

Since  d\  is  between  a  and  b,  bv  between  b  and  c,  &c,  it  is  easy  to  see  (Art. 
,  252)  that  if  we  substitute  successively  al5  bL,  &c,  in  place  of  x  in  F(.r),  the 
results  will  bo  alternately  negative  and  positive  ;  so  that 

For F(o,),  F(M,  F(c,),  &c, 

we  have     ....       — ,       -f-,       — ,    &c. 
But  we  may  apply  to  the  function  F'(.r)  and  its  derived  function  F"(ar)  all 
that  has  been  said  in  the  preceding  article  of  F(.r)  and  F'(.r) ;  then, 

For  ....  F"(a,),  F"(6i).  F"(r')'  &c' 
we  have.  .       +>  — ,        -{-,      &c. 

Thou  the  products  F(a1)xF"(aJ),  F^JxF"^,),  &c.,  of  which  there  are 
m — i,  will  be  all  negative. 

But  if  we  make  F(r)  xF"(x)=y,  and  eliminate  ^as  at  p.  157)  x  between 
the  two  equations, 

F'(.*)=0,  F(x)xF"(x)=y (2) 

the  m  —  1  roots  of  the  final  equation  in  y  will  be  precisely  tho  products  above; 
but  since  all  these  products  are  negative,  the  equation  in  y  will  have  only 
negative  roots,  and,  consequently,  all  its  terms  will  have  tho  sign  -f--  Thus, 
when  the  equation  F(.r)=0  has  none  but  real  and  unequal  roots,  the  theorem 
of  Rolle  shows  that  the  roots  of  F'(.i)=r:0  must  be  real  and  unequal  also;  and 
from  what  has  just  been  said  above,  it  appears  that  besides  this,  the  signs  are 
all  plus  in  the  equation  in  y,  resulting  from  the  elimination  of  x  between  the 
equations  (2). 

262.  Conversely,  these  conditions  being  fulfilled,  we  can  demonstrate  that 
all  the  roots  of  F(.r)  =  0  will  be  real  and  unequal.  And  first,  the  m  —  1  roots 
of  F'(-r)=0  being  real,  from  what  has  just  been  said,  those  of  F"(.r)=0  must 
he  real,  and  the  m  — 1  values  of  y,  or  F(.r)  xF"(.r)  real  also;  and  the  roots 
of  F'(.r)=0  being  by  hypothesis  unequal,  the  theorem  of  Rolle  proves  that  the 
quantities  F"(«i),  F"(&i),  £cc,  have  their  signs  alternately  -f-  and  — .  Again, 
since  the  equation  in  y  has  its  signs  all  +'  we  conclude  that  it  has  no  positive 
roots ;  and  since  all  its  roots  are  real,  they  can  only  be  negative ;  then  the 
m  —  l  products 

V(ai)xF"(a1),F(bl)xF"(b1),6cc, 
are  negative.     But  the  second  factors  have  their  signs  alternate'}-  -(-  and  — 
then  the  quantities  F(a1),  F(5,),  &c,  must  have  their  signs  alternately  —  and 
■•\-.     Then  there  exists  above  «i  a  root  of  the  equation  F(.r)  =  0,  another  be-, 
tween  ax  and  bu  another  betwTeen  bx  and  Cj,  &c,  therefore  tho  m  roots  of  this 
equation  are  real  and  unequal. 

The  conditions  drawn  from  the  equation  in  y  may  be  regarded  as  actually 
known,  because  this  equation  is  obtained  by  simple  elimination.     As  to  the 


J44  ALGEBRA. 

other  condition  which  requires  that  the  roots  of  F'(.r)=0  be  real,  let  it  be  ob 
served  that  this  equation  is  of  the  degree  m  —  1,  and,  applying  to  it  the  same 
reasoning  as  to  F(x)=0,  we  reduce  the  question  to  determining  the  reality  of 
the  roots  of  F"(x)=0,  which  is  only  of  the  degree  m — 2.     Continuing  thus, 
we  descend  to  an  equation  of  the  second  degree,  the  derived  function  of  which 
jeing  of  the  tirst  degree,  can  not  have  an  imaginary  root.    Then  fho  only  con 
dition  to  fulfill  will  be  that  the  equation  y,  which  is  also  of  the  first  degree 
have  its  two  terms  of  the  same  sign. 

Remark. — By  recurring  to  the  reasoning  which  led  to  the  use  of  the  equa 
tion  y  =  F(r)xV"(.i),  it  is  easily  perceived  that  this  may  be  replaced  by 
M  x  F(r)  X  F"(x),  M  being  any  positive  quantity  whatever.    We  can  then  in- 
troduce or  suppress  in  the  polynomials  F(x),  F'(x),  F"(.r),  &c,  such  positive 
factors  as  may  be  judged  suitable  to  simplify  the  calculation. 

263.  The  equation  in  y,  resulting  from  the  elimination  of  x  in  the  equations  (2», 
being  of  tho  degree  m  —  1,  will  have  m —  1  coefficients,  thus  presenting  m — 1 
conditions  to  be  fulfilled;  the  second  equation  in  y,  obtained  by  eliminating  x 
between  the  two,  F"(a:)  =  0,  y  =  F'(.r)  X  F'"(.r),  will  be  of  the  degree  m — 2, 
and  present  m — 2  conditions  to  be  fulfilled,  and  so  on,  till  we  arrive  at  an  equa- 
tion of  tho  first  degree  in  y,  which  will  give  but  a  single  condition ;  then, 
taking  all  the  conditions  in  au  inverse  order,  their  number  will  be  express 
ed  (Art.  228)  by  . 

m(m  —  1) 
1+2+3 |-ro— 1=-^— — -. 

264.  For  an  application  of  the  above,  let  us  take  the  general  equation  of  tho 
second  degree, 

x9+px+f=0. 

Here  we  have  F(x)=x*+px+<jr,  F'(x)=2x-\-p,  F"(.r)  =  2,  and  we  per 
ceive  at  once  that  F'(x)  has  no  imaginary  root,  since  it  is  of  the  tirst  degree. 

In  order  to  have  the  equation  in  //.  the  two  equations  between  which  we 
must  eliminate  x  are 

2x+p  =  0,  y  =  {.i-+F.u+q)  X  2. 

The  elimination  gives 


y+ 


2(\p>-q)=0. 


Then,  in  order  that  the  terms  of  this  equation  may  have  the  same  sign,  we 

1 
must  have  -p'1 —  <7>0;  and  this  is  the  only  condition  necessary  to  insure  the 

reality  of  tho  roots  of  the  equation  of  the  second  degree.     It  accords  with 
what  we  have  seen  at  (Art.  191). 

265.  Let  us  consider  next  tho  general  equation  of  the  third  degree.  The 
second  term,  it  will  be  seen  hereafter,  may  be  made  to  disappear  without 
changing  the  number  of  the  real  roots;  we  may  therefore  take  it  under  tho 
form 

x3+9X+r=0. 

In  this  case  F(x)=x»+gx+r,  F'(x)=3x«+?,  F"(x)=6x.     It  is  oeceei 

[fisl.  thai  the  derived  (•(iiiation.  :: ./-'+ q  —  (I,  should  have  only  real  and  unequal 
roots;  and  for  this  the  condition  is  evidently  '/<!). 


METHOD  OF  FOURIER.  345 

Secondly,  it  is  necessary  to  eliminate  x  between  the  two  equations 
3.r2+?=0 (1) 

7/  =  (.r3+«7.r+r)x6.r, 


or 


The  first  gives 
and  (2)  becomes 


y  —  G.ri-{-r,IJ.r--\-Gr:r (2) 


*= -3?  ■•■*= j 


y=—-q*+6rx 


3y+4q* 

Substituting  this  in  (1),  we  have,  after  reducing, 

2/9+3<?3?+9?(4$3+27r*):=0. 

In  order  that  the  three  terms  of  this  equation  may  have  the  same  sign,  it  is 
necessary,  and  it  is  sufficient,  that  the  known  term  should  be  positive.  We 
have  already  seen  that  g  must  be  negative,  but  q"  in  the  second  term  is  posi 
tive  ;  then  the  new  condition  is  4q:i-{-27r-<^0.  Finally,  as  this  new  condition 
can  lie  fulfilled  only  when  q  is  negative,  it  is  the  only  one  necessary,  in  order 
that  the  roots  of  the  equation  of  the  third  degree  should  be  real  and  unequal. 

FOURIER'S  METHOD  OF  SEPARATING  THE  ROOTS. 

266.  We  shall  now  give  another  method  of  separating  the  roots,  proposed 
by  Fourier,  which  has  the  recommendation  that  the  auxiliary  functions  em- 
ployed in  it  are/(.r)  and  its  successive  derived  functions,  which  can  be  form- 
ed by  inspection  ;*  so  that  the  method  can  be  applied  nearly  with  equal  ease 
to  an  equation  of  any  degree  ;  in  particular,  the  intervals  in  which  no  real  root 
can  be  situated  are,  by  Fourier's  method,  immediately  assigned.  The  objec- 
tion to  this  method  is,  that  by  its  immediate  application  we  only  find  a  limit 
which  the  number  of  real  roots  in  a  given  interval  can  not  exceed,  and  not  the 
absolute  number;  and  that  the  subsidiary  propositions  by  which  this  defect  is 
supplied  are  not  of  the  same  simple  character  as  the  origin1*!  theorem.  The 
enunciation  and  proof  are  as  follows. 

THEOREM. 

The  number  of  real  roots  o/"f(x)=0  which  lie  between  two  timbers  a  and  b, 
can  not  exceed  the  difference  between  the  number  of  variations  *f  signs  in  the 
residts  of  the  substitutions  of  a  and  b  for  x,  in  the  se~>es  formed  &*.  i'(x)  and  its 
derived  functions  :  viz.,/(z),  f'(x),f"(x),  .  ..f"(.r). 

If  none  of  the  equations 

f(x)=0,f'(x)x=0,  &c, 
have  a  root  between  a  and  b,  it  is  manifest  that  the  substitution  of  a  and  b,  and 
of  any  intermediate  quantity,  in/(.r),  f'(x),  &c,  will  always  produce  exactly 

*  Tin-  in.  tliod  of  Sturm  employs  only  the  given  and  first  derived  function  f{x)  pw)  f'(x), 
which  are  the  same  as  V  and  \'u  the  other  functions  in  his  method,  viz.,  Y ..  Y ..  \r.  be- 
,i    obtained  by  the  method  of  the  common  divisor,  which,  in  pra  tedious  forfixuo 

lions  of  the  higher  degrees,  especiallyif  they  have  large  coefficients.    For  t'sim- 

plifying  these  Laborious  opera'aons,  see  Young's  Tin  ory  and  Solution  oft]  Equations 


346  ALGEBRA. 

the  same  series  of  signs;  but  if  any  of  these  equations  have  roots  between  a 
and  6,  then  changes  in  the  will  occur  in  substituting  gradually 

ascending  quantities  from  a  to  b  ;  our  object  is  to  show  that  by  such  substitu- 
tions  the  number  of  variations  of  signs  can  never  increase,  and  that  one  varia- 
tion will  be  lost  every  time  the  substituted  quantity  passes  through  a  real  root 
/(r)  =  0;  this  we  shall  do  by  examining  separately  each  of  the  cases  in 
which  the  series  of  signs  can  be  affected  ;  namely,  1,  when  /('")  Blone 
vanishes;  2,  when  some,  derived  function,  f'"{x),  alone  vanishes;  3  and  4, 
when  some  group  of  derived  functions,  of  which  /'( ')  either  is  nut  or  is  a 
part,  alone  vanishes;  and  lastly,  when  several  or  all  of  these  cases  of  vanish- 
ing happen  at  the  same  time. 

First,  suppose  that  x=c  (c  being  some  quantity  between  a  and  b)  makes 
J'(.r)  vanish,  without  making  any  of  the  derived  functions  vanish;  then  the 
result  of  substituting  c-\- h  for  x  in/(.r)  and f'{x)  is  (supposing  h  so  small  that 
the  signs  of  the  whole  of  the  two  series  which  express  f(c-\-h)  and  f'(e-\-h) 
depend  upon  those  of  their  first  terms,  and  writing  down  only  the  first  terms) 

/i. /'(c)  and /'(c), 

which  have  different  or  the  same  signs  according  as  h  is  —  or  -f-  ;  therefore, 
in  passing  from  c  —  h  to  c-\-h  through  a  root  of  the  equation,  a  variation  of 
signs  is  lost,  but  none  gained.  • 

Secondly,  suppose  that  x=c  makes  one  of  the  derived  functions,  /In(j), 
vanish,  without  making  any  other  of  the  derived  functions,  or/(.r),  vanish  ;  then 
the  result  of  substituting  c-j-/i  for  x  in  the  three  consecutive  functions 

/-'(.r),  /'"(.r),  /"+.(,-), 
(these  being  the  only  terms  which  it  is  necessary  to  examine)*  is 

/-*(<:),  h.f°+i(c),f°+i{c). 

If,  then,  the  first  and  third  terms  have  the  same  sign,  there  will  be  two  varia- 
tions when  h  is  negative,   and  two  permanences  when  It  is  positive;  if  the 
extreme  terms  have  contrary  signs,  there  will  bo  one  variation,  and  one  only 
whether  h  be  negative  or  positive;  therefore,  in   |  from  c  —  h  to  c-\-h 

i  a  value  which  makes  one  of  the  derived  functions  vanish,  either  two 
variations  or  none  will  be  lost,  but  none  ( 

Thirdly,  suppose  that  z=c   I  r  consecutive  di  rived  functions  vanish, 

without  making  any  other  derived  function,  or/(.c),  vanish;  then  the  result  of 
the  substitution  of  c-\-h  for  x  in  the  sei . 

/—r(.r),  /'«-'+'(.r),  .../•»-«(.«■),  /-(*),  /"'+1(<). 
(these  being  the  only  terms  necessary  to  be  examined)  is 

/,n-r(<0> |77",+1(<-).  •  •  ••  |T/,n+'('-)-  y/"'+1(<-).  /m+1(0. 

where  \r  denotes  1.2.3..../'. 

If,  then,  the  extremes  of  this  series  have  the  same  Blgn,  there  will  be  r  oi 
r-|-l    changes   (according   as    r   is  even   or  odd)   when   /;    is    ni  and   no 

when  h  is  positive;    if  the  extreme  terms  have  contrary  signs,  t' 

*  " 

*  It  i-   ;  attei  d  to  the  other  functions  of  tb  of  derived  fanotions,  be- 

. - : i  .  pposed  so  small  tint  net  one  i     I  hea  bj  the  substitution  of  any 

;  i      bet     •    i   c — k  ami  c-\  h,  end  tJ 
--i-h. 


METHOD  OF  FOURIER.  34? 

will  be  r  or  r-j-1  variations  (according  as  r  is  odd  or  even)  when  h  is  negative, 
and  one  change  when  h  is  positive;  therefore,  in  passing  from  c — h  to  c-j-A 
through  a  value  which  makes  r  consecutive  derived  functions  vanish,  r  or  rJtl 
changes  are  lost  (according  as  r-is  even  or  odd)  but  none  ever  gained. 

Fourthly,  suppose  the  vanishing  group  to  consist  of  f(x)  and  the  first  r — 1 
derived  functions  (which  corresponds  to  r  roots  =c  in  J'(.r)=0)  ;*  then  the  re 
suit  of  the  substitution  of  c  +  /t  for  x  mf(x),f'(x),  . . . /^(x),  f*{x),  is 

j7/r(c)'f^/r('')---Y/r(c')'/r(c)' 

in  which  there  are  r  variations  when  It  is  negative,  and  none  when  h  is  posl 
five  ;  therefore,  in  passing  through  a  root  which  occurs  r  times  in  the  equation, 
r  changes  are  lost,  but  none  gained. 

Lastly,  suppose  the  substitution  of  x=c  to  produce  several,  or  all  of  the 
above  cases  at  the  same  time ;  then,  because  the  conclusions  respecting  the 
effect  of  the  passage  through  c  upon  the  series  of  signs  in  ono  part  of  the 
series  of  derived  functions  are  not  at  all  influenced  by  what  happens,  in  con- 
sequence of  the  same  passage,  at  another  distinct  part  of  the  series,  by  what 
has  been  proved,  several  variations  will  be  lost,  but  none  ever  gained. 

Since  then,  in  substituting  gradually  ascending  values  from  a  to  b,  variations 
of  signs  are  generally  lost  for  every  passage  through  a  quantity  which  makes 
one  or  more  of  the  derived  functions  vanish,  and  invariably  one  for  every  pass- 
age through  a  root  of  f(x)=0,  but  none  under  any  circumstances  gained,  it 
follows  that  tho  numler  of  roots  of/(;r)  =  0,  which  lie  between  a  and  b,  can 
not  be  greater  than  the  excess  of  the  number  of  variations  given  by  x=a,  above 
that  given  by  x=&. 

267.  Hence,  if  the  li.nits,  a  and  b,  be  — co  and  -j-00'  or  any  two  numbers 
the  first  of  which  gives  only  variations,  and  the  second  only  permanences ;  and 
if,  in  the  series  formed  by/(.r)  and  its  derived  functions, 

c  be  substituted  for  x  and  be  then  made  to  assume  all  values  between  these 
limits,  the  series  of  signs  of  the  results  will  have  the  following  properties ; 
there  will  at  first  be  n  variations  of  sign,  and  at  last  no  variation,  but  n  per- 
manences; these  variations  disappear  gradually  as  c  increases,  and  when  once 
lost,  can  never  bo  recovered  ;  one  variation  disappears  every  time  c  passes 
through  a  real  unequal  root  of/(.r)=0;  r  variations  disappear  every  time  c 
passes  through  a  root  which  occurs  r  times  in/(.r)=0  ;  either  two  or  none  of 
the  variations  disappear  every  time  one  only  of  the  derived  functions  vanishes, 
without  f{x)  vanishing  at  the  same  time;  an  even  number  p  of  variations  dis- 
appears every  time  an  even  group  of  p  functions  (not  including  the  first  f{x)) 
vanishes;  and  an  even  number  q^zl  of  variations  disappears  every  time  an 
odd  group  of  q  functions  (not  including  the  first/(.r))  vanishes.  Also,  if  a  value 
causes /(.r)  and  the  first  r — 1  derived  functions  to  vanish,  and  an  even  group 
of  p  functions  in  one  part  of  the  series,  and  an  odd  group  of  q  functions  in  an- 
other part,  to  vanish  at  the  same  time,  the  number  of  variations  lost  in  pass 
ing  through  that  value  will  be  r-\-p-\-q±l. 

268.  Hence,  if  f(x)  =  Q  have  all  its  roots  real,  no  value  of  x  can  make  any 
of  the  derived  functions  vanish,  and  thereby  exterminate  variations  of  signs 

*  See  (Art.  253,  Schol). 


34ri  ALGEBRA. 

without  .it  the  same  time  making /(x)  vanish;  for  if  it  could,  since  tliose  vari- 
ations can  never  be  testored,  and  since  b  variation  must  disappear  for  ev< 
passage  through  a  real  root,  the  total  number  of  variations  lost  would  surpass 
/'.  tin-  degree  of  the  equation,  which  is  absurd,  since  there  are  but  n  derived 
functions  in  all.  Whenever,  therefore,  variations  disappear  between  vnluesof 
X  which  do  not  include  a  root  of/(x)=0,  there  is.  correspond  Dg  to  thai  oc- 
currence, an  equal  number  of  imaginary  roots  of /'(.;)  =  0.  He  ce,  if  .r=c 
produces  a  zero  between  two  similar  signs,  or  if  it  produces  an  e\  m  i  hi  i  er 
p  of  consecutive  zeros  either  between  similar  or  contrary  signs,  tl  U  be 

respectively  two,  or  p,  imaginary  roots  corresponding;  or  if  it  produces  an 
odd  number  q  of  consecutive  zeros,  there  will  be  q^zl  imaginary  i 
sponding,  according  as  they  stand  between  similar  or  contrary  signs  ;  <■.  of 
course,  not  being  a  root  of /(.r)  =  (l. 

Observation. — Since  the  derivatives  which  follow  any  one  fT(.c)  ma 
supposed  to  arise  originally  from  it,  it  is  manifest  that  the  same  conciue  •     - 
respecting  the  roots  of/r(.r)  =  0  may  be  drawn  from  observing  the  part  of  the 
series  of  derived  functions 

f'(x),f'+>(x),...f°(z) 

as  were  drawn  respecting  the  root  of/(x)  =  0from  the  whole  series. 

269.  Des  Cartes's  rule  of  signs  is  included  in  Fourier's  theorem  as  a  par 
ticular  case. 

For  when,  in  the  series  formed  by /(.<■)  and  its  derived  functions,  we  put 
•T=  —  cc,  there  are?i  variations  ;  and  when  we  put  .r=0,  the  signs  of  the  series 
of  functions  become  the  same  as  those  of  the  coefficients  of  the  proposed  equa- 
tion 

P^jPa-U  •••Pi,   1. 

Let  the  number  of  variations  in  this  series  of  coefficients  =/,-,  and  there!" 
the  number  of  permanences  (supposing  the  equation  complete)  =n  —  k  ;  if 
we  make  ./ =  +  ao,  the  signs  of  the  functions  are  all  positive,  and  the  number 
of  variations  =0.  Hence,  between  x=  — oo  and  ./=(),  the  number  of  varia- 
tions lost  is  7i— k;  therefore  in  a  complete  equation  there  can  not  be  more 
than  n—k  negative  roots,  i.  e.,  than  the  number  of  permao  i  the  set 

of  coefficients ;  also,  between  x  =  0  and  x=  ,  .  the  number  of  variations  lost  is 
Tc,  whether  the  equation  be  complete  or  incomplete:  hence  in  any  equation 
there  can  not  be  more  positive  roots  than/.-,  i.e.,  than  the  number  of  variation 
in  the  series  of  coefficients,  which  is  Des  Cartes's  rule  ofsiens. 

270.  Fourier's  theorem  may  also  be  presented  under  the  following  form  : 
If  an  equation  have  m  real  roots  between  a  and  }>,  then  the  equation  whose 

roots  are  those  of  the  proposed,  each  diminished   by  a,  has   at    least    m   more 

variations  of  signs  than  the  equation  whose  roots  are  those  of  the  proposed,  each 
diminished  by  b. 
The  transformed  equations  would  be 

'+a)=0,/(y+o)=0; 
and  if  these  were  arranged  according  to  ascending  powers  of  y,  the  coefficients 

would  be  the  values  assumed  by  /'(r),  /"'(,),  ,\  ,-..  when  a  and  b  are  respectively 

written  for  x.     Therefore,  whatever  number  of  variations  of  signs  is  lost  in  tho 

series  /'(,•),/'(,/■),  cVc,  in  passing  from  ,i  to  /-.  the  same  is  lost  in  passing  from 
one  transformed  equation   to  the  other;   but  the   series   for  a   has   at    least   m 


METHOD  OF  FOUIUER.  349 

more  variations  than  that  for  b  ;  therefore  /(?/  + a)  =  0  has  at  least  m  more 
variations  than  j\i/-\-b)  =  0. 

271.  To  apply  this  method  to  find  the  intervals  in  which  the  roots  of 
f(x)  =  0  are  to  be  sought,  we  must  substitute  successively  for  .r,  in  the  series 
formed  by/(>)  and  its  derived  functions,  the  numbers 

_„   ..._10,  —1,0,  1,  10,...,  +/3  (1), 

I  a  and  -\-(i  being  the  least  negative  and  least  positive  number,  which  lmvo 
respectively  only  variations  and  permanences),  and  observe  the  number  of 
variations  of  sign  in  each  result. 

Let  h  and  k  be  the  numbers  of  variations  of  sign  when  any  two  consecutive 
terras  in  series  (1),  a  and  b,  are  respectively  written  for  x ;  therefore  h—k  is 
the  number  of  real  roots  that  may  lie  between  a  and  b  :  if  this  equals  zero, 
f(x)=0  has  no  real  root  between  a  and  6,  and  the  interval  is  excluded;  if 
h—k  =  l,  or  any  odd  number,  there  is  at  least  one  real  root  between  a  and  b  ; 
\l'h—k  =  2,  or  any  even  number,  there  may  be  two,  or  some  even  number,  or 
none  ;  the  latter  case  will  happen  when,  as  explained  above  (Art.  268),  some 
number  between  a  and  b  makes  two  or  some  even  number  of  variations  vanish, 
without  satisfying/^) =0.  Similarly,  we  must  examine  all  the  other  partial 
intervals ;  and  when  two  or  more  roots  are  indicated  as  lying  in  any  interval, 
their  nature  must  be  determined  by  a  succeeding  proposition. 

The  two  former  of  the  following  examples  are  extracted  from  Fourier's 
work. 

EXAMPLE  I. 

/    (x)=       .c6—   3x*—  24.x-3 -f   95.r2—46.r— 101  =  0 

/'  (,r)  =     5x4— 12.T3—   72x3+190x  —  46 

f"{x)  =  20.r;  —  36x2—  i  n.r  +190 

f'"(x)=   60.i:  —  72.C  —144 

f*  (x)  =  120.f  —72 

P  (x)  =  120. 

Hence  we  have  the  following  series  of  signs  resulting  from  the  substitutions 
of  — 10,  — 1,  0,  &c.,  for  x,  in  the  series  of  quantities 

j  j,  J„  jn, 

(-10)     -        +        -        + 

(-1)     +       -       +       - 
(0)  -         -        + 

(i)         -       +       +-      - 

(10)        +        +        +        + 

Hence  all  the  roots  lie  between  —10  and  -\-10,  because  five  variations  have 
disappeared;  one  root  lies  in  each  of  the  intervals  — 10  to  — 1,  and  — 1  to  0, 
because  in  each  of  them  a  single  variation  is  lost ;  no  root  lies  between  0  and  1, 
because  no  variation  is  lost  between  those  limits;  and  three  roots  may  be  sought 
between  1  and  10  (because  three  variations  have  disappeared),  one  of  which  is 
certainly  real ;  it  is  doubtful  whether  the  other  two  are  real  or  imaginary. 

Observation.—  -When  any  value  c  of  x  makes  one  of  the  derived  func- 
tions,/'"^), vanish,  we  may  substitute  c^zh  instead  of  c,  h  being  indefinitely 
small;  then  all  the  other  functions  will  have  the  same  sign  as  when  x=c,  and 
the  sign  of /'"(c i/')  will  depend  upon  that  <  f  ±//m+1(e);  i.  e.,  it  will  be  the 


r 

r 

— 

+ 

— 

+ 

— 

+ 

+ 

+ 

+ 

+ 

350  ALGEBRA. 

same  or  contrary  to  that  of  the  follo\.  Viitive,/m+1(c),  according  as  h  is 

positive  or  negative,  or  according  us  we  substitute  a  quantity  a  little  less  or  a 
little  greater  than  the  value  which  makes fm(x)  vanish.  The  use  of  this  re- 
mark will  be  seen  in  the  following  example. 

EXAMPLE  II. 

/    (r)=     3*—  4a?— 3a:+23=0 


/'  (*)= 

4.r 

l  —  12.r; 

!  — 3 

f"U)  = 

L2a? 

'—24a: 

/'"(*)= 

24x 

—24 

r  (x)= 

24. 

/ 

/' 

/" 

/"' 

r 

xz=o          + 

— 

0 

— 

+ 

x=0j-h,      + 

— 

± 

— 

+ 

x=l              + 

— 

— 

0 

+ 

x=l=fh,      + 

— 

— 

=F 

+ 

1=10           + 

+ 

+ 

+ 

+ 

Every  value  less  than  0  gives  results  alternately  -\-  and  — ,  therefore  there 
i?  do  real  negative  root;  for  .r=0,  we  have  a  result  zero  placed  between  two 
similar  Bigns,  and  therefore  corresponding  to  it  there  is  a  pair  of  imaginary 
roots.  There  is  no  root  between  0  and  1,  but  there  may  be  two  roots  be- 
tween 1  and  10. 

EXAMPLE  III. 

f(x)=x6— 6.r,+40r«+G0.r!  —  x— 1=0. 

Here  there  is  no  root  <C —  1;  there  is  one,  and  there  may  be  three,  be- 
tween  — 1  and  0;  there  is  one  root  between  0  and  1,  and  there  may  be  two 
roots  between  2  and  3. 

272.  The  above  process  will  determine  the  intervals  in  which  the  roots  are 
to  be  sought,  but  not  always  their  nature ;  when  an  even  number  of  roots  is 
indicated,  they  may  all  turn  out  to  be  impossible.  The  series  of  magnitudes 
between  — co  and  -f-oo,  to  bo  substituted  for  x  in  the  d<  rived  functions,  has 
divided  into  intervals  of  two  sorts,  each  contained  by  a  I  limits,  a 

and  b.  The  first  sort  of  interval  is  one  within  which  no  root  is  comprehended, 
i.  c,  the  limits  of  which  give  the  same  number  of  variations  of  Bigns  in  the 
of  derived  functions.  The  second  sort  is  one  within  which  roots  may 
lie,  i.  e.,  where  the  number  of  variations  resulting  from  the  substitution  of  o 
is  bss  than  the  number  resulting  from  the  substitution  of  a,  in  the  s<  ries  of 
derived  functions.  This  second  sort  of  interval  has  two  subdivisions,  viz., 
cases  where  the  indicated  roots  do  really  exist,  and  others  where  they  are 
imaginary.  When  we  have  ascertained  that  a  certain  number  of  roots  may 
tie  between  a  and  b,  we  may  substitute  c  (a  quantity  between  it  and  l>)  in  the 
Beries  of  dorived  functions,  and  if  any  variations  disappear,  our  interval  is  broken 
into  two  others;   if  no  variations  disappear,  we  may  increase  or  diminish  r,  and 

make  a  second  substitution,  and  it  may  still  happen  that  no  variation  is  lost,  and 

80  on  continually;    and   we    may    be   left,  after  all,  in   a  stqjo  of  uncertainty, 
whether  the  separation  of  the  routs  is  impossible  because  they  are  imaginary. 

or  only  retarded  because  their  difference  is  extremely  small.     This  uncer 
tainty  is  relieved  by  taking  the  interval  so  small  as  to  be  sure  to  include  the 

real  roots,  if  they  exist. 


TRANSFORMATION  OF  EQUATIONS.  351 

One  method  of  arriving  at  the  proper  interval  is  by  means  of  the  so-called 
equation  of  the  squares  of  the  differences  of  the  roots  of  the  given  equation, 
which  we  shall  hereafter  have  occasion  to  deduce.  This  process  is  tedious  in 
practice;  and  as  our  object  in  unfolding  the  method  of  Fourier  was  to  pursue 
it  only  so  far  as  it  threw  light  upon  the  general  theory  of  equations,  we  shall 
here  leave  it. 

We  should  now  introduce  the  theorem  of  Budan,  but  it  requires  a  trans- 
formation which  we  have  not  yet  exhibited,  and  we  therefore  take  this  op 
portunity  to  complete  a  subject,  one  proposition  of  which  (Art.  251)  we  liave 
already  had  occasion  to  anticipate. 

TRANSFORMATION  OF  EQUATIONS. 

PROPOSITION   I. 

273.  To  transform  an  equation  into  another  whose  second  term  shall  be  removed 

Let  the  proposed  equation  be 

arn+Ai2n-14-ABa:n-*+ An_1.r+An=0 ; 

and  by  Art.  245  we  know  that  the  sum  of  the  roots  of  this  equation  is  — A! , 
therefore,  the  sum  of  all  the  roots  must  be  increased  by  A!  in  order  that  the 
transformed  equation  may  want  its  second  term  ;  but  there  are  n  roots,  and 

Ai 
hence  each  root-must  be  increased  by  — ,  and  then  the  changed  equation  will 

have  its  second  term  absent.     If  the  sign  of  the  second  term  of  the  proposed 

equation  be  negative,  then  the  sum  of  all  the  roots  is  -f-Ai ;  and  in  this  case 

Ai 
we  must  evidently  diminish  each  root  by  — ,  and  the  changed  equation  will 

7b 

then  have  its  second  terra  removed.     Hence  this 

KULE. 

Find  the  quotient  of  the  coefficient  of  the  second  term  of  the  equation 
divided  by  the  highest  power  of  the  unknown  quantity,  and  decrease  or  in- 
crease the  roots  of  the  equation  by  this  quotient,  according  as  the  sign  of  the 
second  term  is  negativo  or  positive. 

EXAMPLES. 

(1)  Transform  the  equation  .t3— 6.r2-|-8.r — 2=0  into  another  whose  second 
lerm  shall  be  absent. 

Here  A[  =  —  6,  and  n=3  ;  •••  we  must  diminish  each  root  by  £  or  2 

1   _6  +8  —2  (2 
2—8        0 
^4        0  "^2 
2  —4 


—2   —4 
o 


0 
•'•  V3 — 4y — 2=0  is  the  changed  equation. 
And  since  the  roots  are  diminished,  we  must  have  the  relation  x=.y-^-2. 

(2)  Transform   the    equation   X*  —  16a? — 6.r-f-15=0   into   another   whose 
second  term  shall  be  removed. 

(3)  Transform  the  equation  x?+15x*-\-123?— 20.r2+14.r— 25  =  0  into  an- 
ithi-r  whose  seond  term  shall  be  absent. 


ots%  ALGEBRA. 

(4)  Change  the  equation  r--\-ax-lrb  =  0  into  another  deficient  of  the  second 

term. 

(o)  tb. [uation    ./ ■■■■  +  . /r  +  &.r+c  =  0   into   another  wanting  the 

•  I  term. 

A  NSW] 

(  >)    ,,'■  —gey8— 518i/— 777=0. 

(3)  ■    _;•  y    ^412^—757^4-401=0. 

(4)  :--"T  +  l,  =  ». 


(, i-       \  ab 

(5)  ^(__6)z+_-  y+C=0. 


PROPOSITION   II. 

2?4.    To  transform  an  equation  into  another  whose  roots  shall  be  the  reetpro 
tals  of  the  roots  of  the  proposed,  equation. 

Let  «.in  +  A1x"-1  +  A;.r'1--+ AB_iX+AB=0  be  the  proposed  equa- 
tion, and  put  V=-',  then  x=-  and  by  writing  -  for  x  in  the  proposed  equa- 

1     J     x  y  y 

tion,  multiplying  by  yn,  and  reversing  the  order  of  the  terms,  we  have  the 
equation 

A„2/n+A,  ,!r'  +  A„_;jr:+ A.y-  +  A,</  +  «=0. 

whose  roots  are  the  reciprocals  of  the  roots  of  the  proposed  equation. 

The  transformation  is  then  effected  by  simply  changing  the  order  of  the  co- 
efficients of  the  given  equation. 

Corollary  1. — Hence  an  equation  may  be  transformed  into  another  whose 
roots  shall  be  greater  or  less  than  the  reciprocals  of  the  roots  of  the  proposed 
equation,  simply  by  reversing  the  order  of  the  coefficients,  and  then  proceed- 
ing as  in  the  Proposition  to  Art.  2^3. 

Corollary  2. — If  the  coefficients  of  the  proposed  equation  be  the  same. 
whether  taken  in  reverse  or  direct  order,  then  it  is  evident  that  the  trans- 
formed equation  will  be  the  same,  as  the  original  one  ;  and,  therefore,  the  roots 
of  such  equations  must  be  of  the  form 

»"i»-;  ^r;  r*-?  r->»7-  ^c- 

~i  "a  ^3  '  i 

Corollary  3. — If  the  coefficients  of  an  equation  of  an  odd  degree  be  the 
same  whether  taken  in  direct  or  inverse  order,  but  have  contrary  signs,  then, 
also,  the  roots  of  the  transformed  equation  will  be  the  same  as  the  roots  of  the 
proposed  equation;  for,  ch  is  of  all  the  terms,  the  original  and 

transformed  equations  will  be  identical,  roots  remain  unchanged  when 

the  signs  of  all  the  terms  are  changed.  \nd  this  will  likewise  be  the  case  in 
un  equation  of  an  even  degree,  provided  only  the  middle  term  bo  absent,  in 
order  thai  the  transformed  equation,  with  all  its  signs  changed,  may  be  identical 
with  the  original  equation. 

Equations  whose  coefficients  are  the  same  when  taken  either  in  dire. 
reverse  order,  are,  therefore,  called  recurring  equations,  or,  from  the  form  of 
the  root  3,  r»  ciprocal  equ 

Corollary  4. — It'  the  sign  of  the  last  term  of  a  recurring  equation  of  an  odd 
legr,  ,.  lu.  _J_,  one  of  the  roots  of  such  equation  will  be  —  l  :  and  if  the 


TRANSFORMATION  OP  EQUATIONS.  353 

of  the  last  term  be  — ,  one  root  will  be  -j-1.  For  the  proposed  equation  and 
the  reciprocal  have  one  root,  the  same  in  each,  and  1  is  the  only  quantity 
whoso  reciprocal  is  the  same  quantity ;  hence,  since  each  of  tho  other  root* 
has  the  same  sign  as  its  reciprocal,  tho  product  of  each  root  and  its  reciprocal 
must  be  positive ;  and,  therefore,  the  last  term  of  tho  equation,  being  the 
product  of  all  tho  roots  with  their  signs  changed,  must  have  a  contrary  sign  to 
that  of  the  root  unity. 

Hence  a  recurring  equation  of  an  odd  degreo  may  always  bo  depressed  to 
an  equation  of  the  next  lower  degreo  by  dividing  it  by  .r-j-1,  or  x — 1,  accord 
ing  as  the  sign  of  the  last  term  is  -{-  or  — 

Corollary  5. — A  recurring  equation  of  an  oven  degree  may  always  bo  de- 
pressed to  another  of  half  tho  dimensions.     For  let  the  equation  be 
£2n+A1.T9n-1  +  A2z-n-2+ A2.r3+A1x+1=0; 

dividing  by  .rn,  and  placing  the  first  and  last,  the  second  and  last  but  one,  &c. 
in  inxtaposition,  we  have 

*"'+?+Ai(*"14~t)  + An_x(x+^)+An=0 [2] 

1 
Assume  y=x-{-~,  then  we  have 

1  1 

x+x=y  •'• x  +x  —y 


.T-+-+2  x*+-=y*-2 


(*+l)4=-r,+^+4(-T2+^)+6  ^,+^=y4-4(y-2)-6 

&c.  &c.  &c.  =y4 — 4?/2-f-2; 


substituting  these  values  of 


*+-.  *?+&  •  •  •  *n+^  in  [2] 

the  resulting  equation  is  of  the  form 

2/»+B1i/»-1+B22/n-2+ Bn_l2/+Bn=0; 

and  tho  original  equation  is  reduced  to  an  equation  of  half  the  dimensions. 

EXAMPLES. 

(1)  Transform  the  equation  x3 — 7ar-f-7=0  into  another  whose  roots  shall 
be  less  than  the  reciprocals  of  those  of  tho  given  equation  by  unity. 

7  —7  +0  +1  (1 
7    _0        0 

"~ o  ~~ o  ~T 

7        7 


7 

7 


14 

.,  7z3+14;2-r-7r4-l=0  is  the  equation  sought,  where  ;-r-l=-,  or  x== 


x1  2+1 

Z 


354  ALGEBRA. 

(2)  Find  the  roots  of  the  recurring  equation 

x6_Gx^+5r5+5xJ— 6x+l=0. 
By  Cor.  4,  this  equation  has  one  root  x=  —  1,  and  the  depressed  e^  uation  i» 

a*— 7a«+ 1  ±l-2 — 7.r  4. 1  =  0. 
Divide  by  x2,  and  arrange  the  terms  as  in  Cor.  5  ;  then 

^+7i-7(r+x)  +  12=()-  •  -(A) 

1  ut  x+-=r  ;  then  x2+-=:2  — 2  ;  hence,  by  substitution,  (A)  becomes 
x  •*, 

22_O_7r_|_lO_0; 

or  z2— 72  +  10=0; 

and,  resolving  the  quadratic,  we  get 

7  ,     /49 

2=2±Vt-10 

7  +  3 
=~2~~ 
=5,  or  2=2. 

Hence  x+-=5,  and  x+-=2,  and  the  resolution  of  these  two  quadiatus 
x  x 

gives 

x=i(5i  V21)  a°<i  *=+*•  or  +1, 

and  the  five  roots  are 

-1,  +1,  +1,  -^ ,  and - ; 

where  >JZd*  Jg-Va^+yg.    25-2j,=-A=)  whkh  is  the. 
2  2  5+V21      2(5+ V21)      5+^21 

5+ V21 
reciprocal  of  the  root - . 

(3)  Give  the  equation  whose  roots  are  the  reciprocals  of  the  roots  of  the 
equation 

tf— 3.x6— 2r,+3x3+12x2+10x— 8=0. 

(4)  Find  the  roots  of  the  recurring  equation 

5T/5_4yi_^3,/3_3^_j_4,/_5_o. 

(5)  Find  the  roots  of  the  recurring  equation 

;rs+ri  +  .r3_}_T:_4_.r_|_  1  =0. 

ANSWERS. 

(3)  8x°— lOx8— 12x«  — 3x3+2x2+3x— 1  =  0. 


l_i_  V—  3    1— V— "5    -3  +  4V-1  -3-4-/-1 

(4)  i,  — r2 — .  — — . 5 .  ™* 5 • 

/_1+V"^3          /_1_V^3               /-1+  -/^3 
(5)    -1,       V -^ ■       V o •        -V 2 ' 

-\ o ' 


and 


TRANSFORMATION  OF  EQUATIONS.  355 

PROPOSITION    III. 

275.  To  transform  an  equation  into  another  whose  roots  shall  be  any  pro- 
posed multiple  or  submultiple  of  the  roots  of  the  given  equation. 

Let  xn-\-  A^-'  +  AaX"-2-! An_,.r4-An=0  be  any  equation  ;  then  putting 

y=mx,  we  have  x=— ,  and  by  substituting  this  value  of  x  in  the  given  equa- 
tion, and  multiplying  each  term  by  ?na,  we  have 

y"-\-mAiya-l-\-m2A.iyu-'2-\ ml,-1An_12/+mnAn=0 ; 

an  equation  whose  roots  are  m  times  those  of  the  proposed  equation.  Hence 
we  have  simply  to  multiply  the  second  term  of  the  given  equation  by  m,  the 
third  by  m?,  the  fourth  by  m3,  and  so  on,  and  the  transformation  is  effected. 

Corollary  1. — If  the  coefficient  of  the  first  term  be  m,  then,  suppressing  m 
in  the  first  term,  making  no  change  in  the  second,  multiplying  the  third  by  m, 
the  fourth  by  m?,  and  so  on,  the  resulting  equation  will  have  its  roots  m  times 
those  of  the  given  equation. 

Corollary  2. — Hence,  if  an  equation  have  fractional  coefficients,  it  may  be 
changed  into  another  having  integral  coefficients,  by  transforming  the  given 
equation  into  another  whose  roots  shall  be  those  of  the  proposed  equation 
multiplied  by  the  product  of  the  denominators  of  the  fractions. 

Corollary  3. — If  the  coefficients  of  the  second,  third,  fourth,  6cc,  terms  of 
an  equation  be  divisible  by  m,  ?«2,  m3,  and  so  on,  respectively,  then  m  is  a  com- 
mon measure  of  the  roots  of  the  equation. 

EXAMPLES. 

(1)  Transform  the  equation  2.T3 — 4x~-\-7x — 3  =  0  into  another  whose  roots 
mall  be  throe  times  those  of  the  proposed  equation. 

(2)  Transform  the  equation  4x* — 3x3 — 12x2-}-5x — 1=0  into  another  whose 
roots  shall  bo  four  times  those  of  the  given  equation. 

1         1 

(3)  Transform  the  equation  aP+^a? — -x-f-2  =  0  into  another  whose  roots 

shall  be  12  times  those  of  the  given  equation. 

ANSWERS. 

(1)  2x!  —  12x2+63x— 81  =  0. 

(2)  x*— 3X3— 48x2-f-80x— 64=0. 

(3)  r34-4x2— 36x+3456  =  0. 

PROPOSITION   IV. 

276.  To  transform  an  equation  into  another  whose  roots  shall  be  the  square* 
of  the  roots  of  the  proposed  equation. 

Let  xn4-A1xn_I  +  A2.Tn_2-f- -f-An-iX-f-An=0  be  any  equation  ;  then 

rn — AiXn-1-r-A2a:n_2 — i  An_iX=pA„  =  0  is  the  equation  whose  roots  ar« 

the  roots  of  the  former,  with  contrary  signs  (Prop.  VII.,  Art.  247). 

Let  «i,  a2,  a3,  &c,  be  the  roots  of  the  former  equation,  anc  — a,,  — a2,  — a3, 
5cc,  those  of  the  latter  ;  then  we  have 

(xn+ A2xn^-| )  +  (A^-i-f  A3xn~3-f-  .  .)  =  {x—al)  (x— a:)(x— a3) .... 

(xn  +  A2x-B+  . . .  .)_(A1x"-1  +  A3x'-3+  . .  .)  =  (x+a1)(-c+<z*)(j:+«.'«) .. 

Hence,  by  multiplying  these  two  equations,  we  have 
xn+Aslx»-24-...)2— (A1z^-I-r.A3x»-3+  ...)2=(x3— ai2)fx»— a^fx3— a,9) .. 


356  ALGEBRA. 

Or  a*— (Ai9— 2A,)a      :  +  (A:;  —  J  A  ;A.+2A  ,)./-"-•'—  . . .  6cc    s=(a*— a,«j 

(a? — «/)('-'  —  Oa9) by  actually  squaring  and   arranging  according  to  tho 

powers  of  x.     Now,  for  ':  write  y,  and  wo  have 

y°-(\, --  .'  \:)^-'  +  (A,:-2A1A3+'JA,).y^-..  &c,  =0/-Gr)(7/-    . 
(y—a3-) . . . 
•.  2/n  —  (A,2— 2A2)7/n_1  +  (A  ?—  >A,  \  ,+2A4)y0-9— =0  is  an  equation 

whose  roots  are  the  squares  of  the  roots  of  the  given  equation. 

I  KAM? 

(1)  Transform  the  equation  x3+3x: —  ftr — 8  =  0  into  another  whose- roots 
are  the  squares  of  those  of  the  proposed  equation. 

Here  x3 —  6x= — 3x2-|-8  by  transposition,  and  by  squaring  we  have 
a*  _  1 2x* -f- 36I9 = Ox1 — 4  8.r- +  64 
.-.  x6— 21x4+84x2  —  64=0, 
or 

y»—  21y»+84y— 64=0 
is  the  required  equation. 

The  roots  of  the  given  equation  are  — 1,  — 4,  2;  and  those  of  the  trans- 
formed equation  are  1,  4,  16. 

(2)  a*+a*+3a*+16ar+15=0. 
The  transformed  equation  is 

x6+2x,+33x3+23x2+166x— 225=0, 
winch  has  (Art.  259)  only  one  positive  root,  and  therefore  the  proposed  h«* 
only  one  real  root. 

(3)  Transform  the  equation  x3 — x3 — 7x-|-15  =  0. 

4)  Transform  the  equation  x4 — 6x3+5x2+2x — 10=0. 

(5)  Transform  the  equation  x4 — ix3 — 8x-|-32=0. 

(6)  Transform  the  equation  x4 — 3X3 — 15x2+49x — 12=0. 

ANSWERS. 

(3)  y*— 15^+79^—225=0. 

(4)  3/«  —  2Gy»+29y3  —  104y+ 100=0. 

(5)  y*  —  16f  —  64y+ 1024  =  0. 

(6)  y*— 39y3+495y9— 2041y+144==0. 

PROPOSITION    V. 

277.    To  transform  an  equation  into  another  wanting  any  given  term. 

By  recurring  to  the  transformed  equation  in  Art.  251,  note,  in  which  the 
roots  of  the  proposed  &ro  increased  or  diminished  by  a  quantity  represented 
by  r,  it  will  be  seen  that  in  ordor  to  know  what  value  r  must  have  to  make  the 
coefficient  of  any  power  of  x  disappear,  it  is  only  necessary  to  place  the  column 
of  quantities  by  which  that  power  is  multiplied  equal  to  zero,  and  the  result- 
ing equation,  when  resolved,  will  furnish  the  proper  values  of  r.  This  equa- 
tion will  be  of  tho  1°  degree  when  it  is  required  that  the  second  term  shall  dis 
appear,  it  will  bo  of  the  2°  degree  when  the  third  is  to  disappear,  and  so  on. 
The  last  term  can  be  made  to  disappear  only  by  means  of  an  equation  of  the 
same  degreo  as  tho  proposed. 

By  removing  the  second  term  from  a  quadratic  equation,  we  shall  ne  imma- 
•  !  ately  conducted  to  the  well-known  formula  for  its  solution.  Thus,  the  equa- 
tion being 


TRANSFORMATION  OF  EQUATIONS.  ^f/7 


.T*-fAx-j-N=0, 
tne  transformed  in  x'-\-r  will  be 

+  A      +  Ar[=0; 
+N   S 
unci,  that  its  second  term  may  vanish,  we  must  have 

2r+A=0  .-.  r=  — |A, 
which  condition  reduces  the  transformed  to 

.r'*— jA2+N=0 


.-.  x=.r'+r=— iA-J;  V]A'-— N; 
which  is  the  common  formula  for  tho  solution  of  a  quadratic  equation. 

PROPOSITION   VI. 

278.  Tb  transform  an  equation  into  one  ivhose  roots  are  the  squares  of  tne 
i/ijfcrcnccs  of  the  roots  of  the  proposed  equation. 

[f  in  the  given  equation, /(.r)=0,  we  make  x=al-\-y,  ax  being  one  of  the 
rpots,  y  will  be  the  difference  between  ax  and  every  other  root.  If  we  make 
r=ai-\-y,  y  will  be  the  difference  between  a«  and  every  other  root,  and  so  on. 

But  since  a^  a2,  &c,  are  rooto  of/(x)  =  0,  they  must  satisfy  it ;  hence 
f(ai)  =  0,f(a,)  =  0,  &c (1) 

If  wo  eliminate  ax  or  a2i  &c,  between  either  of  these  equations  (1)  and  the 
corresponding  ones,  f{ax-\-y)  =  0,  f(a.t+y)=:0,  &c,  tho  final  equation  in  y 
will  bo  in  each  case  the  same,  and  is  therefore  the  equation  whose  roots  are 
the  differences  of  tho  roots  of  the  proposed  equation.  It  is  evidently  the  same 
thing  to  eliminate  between /(.r)  nm\f(x-\-y). 

The  form  of  tho  equation  f{r-\-y)  is  (Art.  251), 

f{x)+Mx)y+r%f+ r- 

The  first  term  is  identical  with  the  proposed  equation,  and  vanishes,  and  the 
whole  is  divisible  by  y ;  we  thus  deduce 

m+f£%y+  •  •  •  r-1 (2) 

The  equation  (2)  is  of  the  ra  —  1  degree,  and  by  elimination  with  the  pro- 
posed equation  of  the  degree  m  will  produce  a  final  equation  of  the  degree 
m(m  —  1),  as  will  be  hereafter  shown.  It  is  evident,  indeed,  that  the  roots 
being  of  the  form  ax— a8,  a2—au  ai—a3,  a3—au  a.2—an,  &c,  will  be  equal  in 
number  to  the  permutations  of  m  letters,  two  and  two,  which  is  m{m—  1) 
(Art.  200).  The  factors  m  and  m  —  1  will  the  one  be  even  and  the  other  odd, 
and  the  product  m(m — 1)  must  therefore  necessarily  be  even  ;  moreover,  since 
if  one  root,  ai—a:,  be  represented  by  /?,  another,  a2— rti,  will  be  represented 
by  — /?,  and  the  equation  (2)  will  be  composed  of  factors  of  the  form  (y  — 
(y-\-j3)=y'i — /?-;  and  hence  will  contain  only  even  powers  of  y.  It  maj 
therefore  be  written  under  the  form 

y*»+pyim-*+qy*m-4^.y  &Ci)   _|_^  —  0     .    .    .    .    (3)       . 

and  if  we  make  y"---~,  we  have 

2m+_p:m-1+ry;m-'"+,  ecc.  +<  =  0 (4) 


358  ALGEBRA. 

as  the  equation  whose  roots  are  the  squares  o*  the  differences  of  the  roots  of 
die  proposed  equation. 

I.  As  an  application  of  the  foregoing  princi]  fes,  let  us  find  the  equation  of 
the  squares  of  the  differences  for  the  equation  of  the  third  degree.  In  tin- 
first  place,  1  snail  make  the  general  remark,  that  equations  (3)  and  (4)  ought 
not  to  change  when  we  augment,  or  when  we  diminish,  by  the  same  quantity 
all  the  roots  of  equation  (1).  Consequently,  if  the  second  term  of  a  giveD 
equation  be  not  wanting,  we  can  cause  it  to  disappear  (Art.  273),  and  then 
find  the  equation  of  the  differences  for  the  transformed  equation ;  we  shall 
thus  find  the  same  equation  as  if  we  had  not  made  the  second  term  vanish,  since 
the  differences  of  the  roots  will  be  the  same  as  before,  while  the  calculations 
will  be  less  complicated.  This  being  premised,  I  will  suppose  that  the  equa- 
tion of  the  third  degree  wants  its  second  term,  and  has  the  form 

r>+qx+r  =  0 [AJ 

Designate  the  given  equation  by/(.r)=0,  and  the  derived  polynomials  of 

f(x)  byj\(x),fi(x)tf3(.v) ;  the  rule  for  finding  the  equation  of  the  squares 

of  the  differences  is  to  eliminate  between  the  two  equations 

/(.r)  =  0,/1(.r)  +  ^(,)-/  +  T^/;(.r)/4-  .  .  .'  =0 [BJ 

But  in  the  case  before  us  we  have 

f(x)=i*+qx+T,  fl{x)=3afl+q,  f;(x)  =  6x,  f3(x)=6. 
Substituting,  therefore,  these  values  in  equations  [B],  we  shall  readily  perceive 
that  the  elimination  of  x  ought  to  be  performed  between  equation  [A]  and  the 
fo Flowing  equation, 

3x*+q+3xy  +  y*=0 [C] 

We  shall,  therefore,  arrange  this  equation  with  reference  to  x,  and  then  elimi- 
nate x  by  proceeding  as  if  we  had  to  find  the  greatest  common  divisor  of  equa- 
tions "A]  and  [C]. 

First  Division. 


x^-\-  qv  -4-   r 
3x34-  3qx  +3r 


:^+3v.r-r-7/*+7 


x — y 
+  3r»  +  3>/3*+(?/°+   q)x 

—3yx2—(y*—2q)x+3r 

—  3}/.r:  —  :;//•'.;■ — ,,     —     . 

Second  Division. 


+q)x+f+qy+9r 


y.r +  ::(■■/  +  ;;i-3r) 


3.r-+  3?.r+    tf+q 

6(?/2+7)^4-,;(.'/-'  +  V).V+-,(.'/:  +  V): 
+  6(?/,+7)^+3(?/3+7.V    +3r).r 
tyf+W  -3r)x+2(y'+7)9 
6W+q){tf+qy  -3r)x+4(3f+?)8 
gjy+jT)(g+gy  —3r)x+Z(y*+qy+3r){y*+qy—3r) 
Hr  +  7y-:;(r  +'l!t  +  '>i-)(!i:t+<!!'-*r). 
In  the  laal  division  we  have  multiplied  twice  by  y:-\-<j  in  order  to  render  th* 
divisions  possible,  but  if  we  take  v'-f  v=°.  the  divisor  reduces  to  Sr,  a  quau 
tity  in  general  differing  from  0. 


BUDAN'S  CRITERION.  359 

Making  the  last  remainder  equal  to  zero,  and  performing  the  operations  in- 
dicated, the  equation  of  the  differences  is 

i/-lr6rpf+Dq"y~  +  4qi+27ri  =  0 ; 
taking  y2=z,  the  equation  of  the  squares  of  the  differences  becomes 

za+6gz8+923z-f4g3+27r8=:0. 
For  the  equation  Is  —  7.r+7:=0,  we  have  q== — 7,  r=-J-7;  and  hence  the 
equation  in  z  becomes 

z3— 42z2+441z— 49=0. 

BUDAN'S  CRITERION 
For  determining  the  number  of  imaginary  roots  in  any  equation. 

280.  If  the  real  positive  roots  of  an  equation,  taken  in  the  order  of  their 
magnitudes,  be  au  a2,  as,  a.t an,  where  ax  is  the  smallest,  and  if  we  dimin- 
ish the  roots  of  the  equation  by  a  number  h  greater  than  au  but  less  than  a2, 
then  the  roots  will  be  ax — h,  a2 — h,  «3 — li,  ...aa — h,  and  the  first  of  these 
will  now  be  negative.  Rut  tho  number  of  positive  roots  is  exactly  equal  to 
the  number  of  variations  of  sign  in  the  terms  of  the  equation  when  the  roots 
are  all  real ;  and  as  we  havo  changed  one  positive  root  into  a  negative  one, 
the  transformed  equation  must  have  one  variation  less  than  the  proposed 
equation. 

Again,  by  reducing  all  the  roots  by  Jc,  a  number  greater  than  a2,  but  less 
than  a3,  we  shall  have  two  negative  roots,  «i — Jc,  a2 — Jc,  in  the  transformed 
equation,  and,  therefore,  we  shall  havo  two  variations  of  sign  less  than  in  the 
proposed  equation,  for  two  positive  roots  have  been  reduced  so  as  to  become 
negative  ones.  Hence  it  is  obvious,  that  if  we  reduce  the  roots  by  a  number 
greater  than  an,  all  the  positive  roots  will  become  negative,  and  the  transform- 
ed equation,  having  all  its  roots  negative,  will  have  the  signs  of  all  its  terms 
positive  (Art.  259),'  and  all  the  variations  will  havo  entirely  disappeared. 

We  see,  then,  that  if  the  roots  of  an  equation  be  reduced  until  the  signs  of 
all  the  terms  of  the  transformed  equation  be  -j-»  we  have  employed  a  greater 
number  than  the  greatest  positive  root  of  that  equation ;  and,  therefore,  its 
reciprocal  must  be  less  than  the  smallest  real  root  of  the  reciprocal  equation. 
Now,  if  we  take  the  reciprocal  equation,  and  reduce  its  roots  by  the  reciprocal 
of  the  former  number,  we  should  have  as  many  positive  roots  left  in  this  trans- 
formed reciprocal  equation  as  there  were  positive  roots  in  the  proposed  equa- 
tion, unless  the  equation  has  imaginary  roots  ;  hence  the  number  of  variations 
lost  in  the  former  case  should  bo  exactly  equal  to  the  number  left  in  the  latter, 
when  the  roots  are  all  real ;  and,  consequently,  if  this  condition  be  not  fulfill- 
ed, the  difference  of  these  numbers  indicates  the  number  of  imaginary  roots. 
To  explain  this  reasoning  more  cloarly,  we  shall  suppose  that  an  equation  has 
threb  positive  roots ;  as,  for  instance, '1,  2-5,  and  3.  Now  if  the  roots  of  the 
proposed  equation  be  reduced  by  4,  a  number  greater  than  3,  the  greatest 
positive  root,  tho  three  positive  roots  in  the  original  equation  will  evidently  be 
changed  into  three  negative  ones  in  the  transformed  one,  and  hence  three  va- 
riations must  be  lost.  Again,  the  equation  whose  roots  are  the  reciprocals  of 
the  proposed  equation  must  have  three  positive  roots,  1,  |,  and  \  ;  and  it  is 
evident  that  if  we  reduce  the  roots  of  the  reciprocal  equation  by  j,  the  recip- 
rocal of  the  former  reducing  number  4,  we  shall  not  change  the  character  of 
Khe  three  positive  roots,  because  j  is  less  than  tho  least  of  them,  and  1 — i 


360  ALGEBRA. 

§ — h  3  —  l  are  a^  positive  ;  hence  the  three  variations  introduced  by  the 
three  positive  roots  must  still  be  found  in  the  transformed  reciprocal  equation, 
and,  therefore,  three  variations  aro  left  in  the  latter  transformation,  indicating 
no  imaginary  roots.     The  theorem  may,  therefore,  be  stated  thus  : 

If,  in  transforming  an  equation  by  any  number  r,  there  be  n  variations  lost, 
and  if,  in  transforming  the  reciprocal  equation  by  \  (the  reciprocal  of  r),  there 
be  m  variations  left,  then  there  will  be  at  least  n — m  imaginary  roots  in  the 
interval  0,  r. 

For  there  aro  as  many  positive  roots  in  the  interval  0,  r  of  the  direct  equa- 
tion as  there  are  between  l-  and  -  of  the  reciprocal  equation  ;  hence,  if  n,  tho 
number  of  variations  lost  in  the  transformation  of  the  direct  equation  by  r,  be 
greater  than  m,  the  number  of  variations  left  in  the  transformation  of  the  re- 
ciprocal equation  by  -,  there  will  bo  a  contradiction  with  respect  to  the  charac- 
ter of  a  number  of  the  roots,  equal  to  tho  difference  n  —  m.  Hence  these* 
roots  are  imaginary. 

EXAMPLE. 

Find  the  number  of  imaginary  roots  of  the  equation 

x*—x*-\-2x*+x  —  4  =  0. 


Direct. 
-1   +2  +1 
10       2 

0        2        3 

-4(1 
3 

—  1 

Reciprocal. 

-4  +  1  +  2-1+1  (1 

_  4   _  3   _   1    _o 

_  3   _   1    _  2   — 1 

1        1        3 
13        6 

—  4   —  7   —  8 

—  7   —   8  —10 

1  2 

2  5 

—  4   —11 
—11    —19 

1 

—   4 

3 

—  15 

Here  two  variations  are  lost  in  the  transformation  of  the  direct  equation, 
and  no  variations  aro  left  in  the  transformation  of  the  reciprocal  equation ; 
therefore  this  equation  has  at  least  two  imaginary  roots  ;  and  it  has  only  two, 
for  the  sign  of  the  absolute  term  is  negative,  implying  the  existence  of  two 
real  roots,  tho  one  positivo  and  the  other  negative.  (See  Art.  248,  Pr.  VIII., 
Cor.  5.) 

EXAMPLE. 

To  find,  the  number  and  situation  of  tho  real  roots  of  tho  equation  x*+ar* 
+z2+3x— 100=0  by  Budan's  method. 

If  the  roots  of  this  equation  be  all  real,  tho  permanences  and  variation  indi- 
cate throo  negati>o  roots  and  one  positive  root. 

(1)  To  find  the  positivo  root. 

1  +  1  +  1+   3  —  100(2  1  +  1+    1+    3  —  100(3 


3  +  7  +  17—   6G 


.1  +  13+1-J+   26 


In  the  transformation  by  2,  one  variation  is  left,  and.  in  transforming  I 
there  is  no  variation  left  ;  therefore  the  positive  root  is  between  2  and  3 

(2)  For  tlif  negative  roots. 


THE  LIMITS  OF  THE  ROOTS  OP  EQUATIONS.  361 


Reciprocal  Equation. 
—  100—     3+     1—     1+     1(1 
—  103—102  —  103  —  102 


Direct  Equation. 
1  —  1  +  1  —  3  —  100  (1 
0+1  —  2  —  102 
1  +  2+0 

2+4  signs  all  — 

3 

Hero  two  variations  are  lost  in  iho  direct  transformation,  and  no  variations 
ire  left  in  the  reciprocal  transformation  ;  therefore  the  two  roots  in  the  inter- 
nal 0  and  — 1  are  imaginary. 


1  —  1  +  1—  3  —  100  (3 

o_|_7_|_i8_   4G 


1_1+   l_  3  —  100  (4 
3+13*+49  +   96 


Hence  the  negative  root  is  obviously  situated  between  — 3  and  — 4. 

DEGUA'S  CRITERION. 

281.  In  any  equation,  if  we  have  a  cipher-coefficient,  or  term  wanting,  and 

if  the  cipher-coefficient  bo  situated  between  two  tornis  having  the  same  sign, 

there  will  be  two  imaginary  roots  in  that  equation. 

Let  the  order  of  the  signs  be 

+  +  -0-  + . 

and  for  0  writing  +  or  —  we  have  either 

+  +  -  +  -  + ,or+  + + 

In  the  former  of  these  we  find  two  permanences  and  five  variations,  and  in 
the  latter  we  have  four  permanences  and  only  three  variations  ;  hence,  if  the 
roots  are  all  real,  wo  must,  in  the  former  case,  havo  five  positive  and  two  neg- 
ative roots,  and  in  the  latter,  three  positive  and  your  negative  roots  (Art.  259) ; 
hence  wo  have  two  roots,  both  positive  and  negative,  at  the  same  time,  and, 
therefore,  these  two  roots  can  not  be  real  roots.  These  two  roots,  which  in- 
volve the  absurdity  of  being  both  positive  and  negative  at  the  same  time,  must, 
therefore,  be  imaginary  roots. 

In  nearly  the  same  manner  it  may  be  shown  that 

(1)  If  between  terms  having  like  signs,  2n  or  2n — 1  cipher-coefficients  in- 
tervene, there  will  be  2n  imaginary  roots  indicated  thereby. 

(2)  If  between  terms  having  different  signs,  2/1+1  or  2?i  cipher-coefficients 
intervene,  there  will  be  2«.  imaginary  roots  indicated  thereby. 

EXAMPLE. 

The  equation  x* — r5+6:r3+24  =  0  has  two  imaginary  roots,  for  the  absent 
term  is  preceded  and  succeeded  by  terms  having  like  signs  ;  and  the  equation 
r'  +  l,  having  the  coefficients  1  +  0  +  0  +  1,  has  also  two  imaginary  roots 

EXAMPLES   FOR   PRACTICE. 

(1)  How  many  imaginary  roots  are  in  the  equation 

xs+r5— 2x-3+2.r— 1  =  0  ? 

(2)  Has  the  equation  x* — 2z2+6x+10  =  0  any  imaginary  roots  ? 

THE  LIMITS  OP  THE  ROOTS  OP  EQUATIONS 
282.  The  limits  of  any  group  of  roots  of  an  equation  are  two  quantities  be- 
tween which  the  whole  group  lies;  thus,  +co  and  0  are  limits  of  the  positive 
roots  of  every  equation,  and  0  aud  — oo  of  the  negative  roots.     But  in  practice 
We  are  required  to  assign  much  closer  limits  than  these,  usually  the  two  con- 


362  AlaEBUA. 

gecutive  whole  numbers  between  which  each  root  lies,  so  that  the  inferior 
lirnit  is  the  integral  part  of  the  included  root.  This  may  be  effected  without 
knowing  any  of  the  roots  of  the  equation,  as  will  be  seen  in  the  following  prop- 
ositions.    The  roots  spoken  of  hi  this  section  are  the  real  roots. 

SUPERIOR  AND    INFERIOR  LIMITS   OF   THE   ROOTS. 

283.  The  greatest  negative  coefficient  increased  by  unity  is  a  superior  limit 
of  the  positive  roots  of  an  equation. 

Let  — p  bo  the  greatest  negative  coefficient;  then  any  value  of  x  which 

makes 

xn— p(xa-l-\-xB-*-\- \-x2-{-x-\-l)  positive, 

xn l 

or  x-^^x^+x—H yxz+x+l)^?—- —  ,* 

will,  a  fortiori,  make 

x"+p]xa-l+p2x"-*+ |_pn_lX+p0, 

otf(x)  positive,  because  in  the  latter,  all  the  terms  after  xn  will  not  generally 
be  negative,  and  of  the  negative  terms  not  one  is  greater  than  the  correspond- 
ing term  in  the  former  expression. 

xn  — 1 
Now  the  inequality  xn>_/; is  satisfied,  if 

xn=  or  >x"  ,  or  x— 1=  or  >_»,  or  x=  or  >^;-(-l. 

Since,  therefore,  p + 1  and  every  greater  number,  when  substituted  for  x. 
will  make/(x)  positive,  the  numerical  value  of  the  greatest  negative  coefficient 
Increased  by  unity  is  a  superior  limit  of  the  positive  roots.f 

284.  In  any  equation,  if  £>rxn-r  be  the  first  term  which  is  negative,  and  — p 
the  greatest  negative  coefficient,  1-f-  \/p  is  a  superior  limit  of  the  positive 
roots. 

Any  value  of  x  which  makes 

x">p(xn-r+xn— '+  . . .  +x+l)>/  x_~  , 
will  of  course  make  xn-r-_p1xn_1-r-p2xn_2-l-  . . .  positive. 

Tn_r+1  _  j 

Now  the  inequality_xn^>p — ,  is  satisfied  if 

^-P'xTrp  0T3?~1(x—1)>Pi  or  if  (*— lr-'fa— 1)=  or  >p,  or  (x— 1)  = 

or  >_/?,  or  x=  or  >1+  Vp. 

Since,  therefore,  1-f-  \J  p  and  every  greater  number  gives  a  positive  result, 
1  -|-  V p  is  a  superior  limit. 

This  method  may  bo  employed  when  the  first  term  is  followed  by  one  or 
more  positive  terms. 

EZAMFl 

x*+lla«— 25x— 61i»3. 

Here  r=3,  and  a  limit  of  the  positive  routs  is 

1+  Vo'l,  or  5,  taking  the  next  higher  integer. 

285.  If  each  negative  coefficient,  taken  positively,  be  divided  by  the  sum  of 


*  Sec  (Art.  23).  t  This  is  oommonly  known  as  Maclaurin's  limit. 


SUPERIOR  AND  INFERIOR  LIMITS  OF  THE  ROOTS.  3G3 

all  the  positive  coefficients  which  precede  it,  the  greatest  of  the  fractions  thus 
forned,  increased  by  unity,  is  a  superior  limit  of  the  positive  roots. 
Let  the  equation  be 

xn+j)lxn-l-irp2xa-2+{—2h)xn-3-\-  ■  •  • 
I  ...+(_pr).r"-'+...+pa=0; 

then,  since  (Art.  23), 

pmx™=jJm{x-l){z«>-1+xm-*+  ...  +x+l)+Pm, 
if  we  transform  every  positive  term  by  this  formula,  and  leave  tho  negative 
tevms  in  their  original  form,  we  shall  have 

0  =  (.r-l)x"-1  +  (.r-l).r"--+(.r-l)xn-3+...+.r-l  +  l 

+i>,(x— l)i^+pi(x— l)x»-3+  . . .  +pi(x—l)+pi 

+jp3(x— iy**-*+  •  •  •  +Mx—1)+p* 

— l)3Xn~3 

+  

Now  if  such  a  value  bo  assigned  to  x  that  every  term  is  positive,  that  value 
will  be  the  superior  limit  required  ;  in  the  terms  where  no  negative  coefficient 
enters,  it  is  sufficient  to  have  x>  1  ;  in  the  other  terms,  each  of  which  in- 
volves a  negative  coefficient,  we  must  have 

(l+JPi+JPh)(*— 1)>1* 

(i+Pi+.P2+---+iv-i)(*— i)>Pm  &c, 

or 

*>T+^+1  '    X>l+p,+PX-+Pr-^  ^ 

If,  then,  x  be  taken  equal  to  the  greatest  of  these  fractions  increased  by 
unity,  this  value,  and  every  greater  value,  will  make  f(x)  positive,  and  there- 
fore will  be  a  superior  limit  of  the  positive  roots.  This  method  gives  a  limit 
easily  calculated,  and  generally  not  far-  from  the  truth.* 

EXAMPLES. 

(1)  4X5— 8x4+23x3+105x2— 80x+3  =  0. 

8  80  8^    80        ,        _        8 

The  fractions  are  -  and  4  ■  23  ■  105»  and  i>i3o  >   therefore  -+1  =  3  is  a 

superior  limit. 

(o)  4.r7— 6.r6— 7x5+8.r4+7.r3— 23x2— 22x— 5=0 ; 

here  3  is  a  superior  limit. 

Observation.— ;The  form  of  the  equation  will  often  suggest  artifices,  by 
means  of  which  closer  limits  may  be  determined  than  by  any  of  the  preceding 
methods  ;  thus,  writing  the  equation  of  Example  1  under  the  form 

4ar*(x — 2)  +  23x3+105x(x— —  )  +  3=0, 

we  see  that  x=  or  >2  gives  a  positive  result,  therefore  2  is  a  superior  limit. 
Similarly,  by  writing  the  example  of  Art.  284  under  the  form 


z(is_25)  +  ll 


(*2-n)=0' 


we  see  that  3  is  a  superior  limit. 

We  have  seen  (Art.  248)  that  an  equation  of  an  even  number  of  dimension!* 
with  its  last  term  positive  may  have  no  real  root ;  but  we  shall  now  show  that 

*  This  is  the  method  of  Bret. 


SJ4  ALGEBRA. 

m  any  equation  whatever,  if  the  absolute  term  be  small  compared  wtu  the 
>ther  terms,  there  will  be  at  least  one  real  root  also  very  small. 
286.  In  the  equation 

Po^n+i»^n~'  +  '  &c->  +r—r=0, 
rvhere  r  is  essentially  positive,  and  which  may  represent  any  equation  what- 
ever, if  r<C  „/,   , ,  where  v  is  numerically  the  greatest  coeflicient,  then  there 

4(1  +p) 

is  a  real  positive  root,  <2r. 

By  dividing  by  the  coefficient  of  x,  and  changing  the  signs  of  all  the  terms, 
and  of  all  the  roots,  if  necessary,  eveiy  equation  may  be  reduced  to  the  form 

—  r+x+,  &c.,  -fp1.r—14-po.rn=0   ....  (1) 
where  r  is  essentially  positive ;  let  p  bo  numerically  tho  greatest  coefficient, 
then  any  value  of  x<C\  which  makes 

-r+x>p{a*+&+,  &c,  +*")>       \_x       , 

will  make  the  first  member  of  (1)  positive  ;  and  this  condition  is  fulfilled  by 

«.r3 
-r+x=  or  >— , 

because  1>1 — rn_1,  or 

(l+p)-v°—(I  +  r)x+r=0, 
or 


2(l+iv).r=(l  +  r)-V(l  +  r)— 4r(l+p); 
if,  then,  4r(l-\-p)<C.l,  the  radical  will  have  a  real  value  >r,  and  there  will  be 

for  x  a  real  value  less  than  Q,  which  makes  the  first  member  of  (1 )  posi- 

tive, while  .r=0  makes  it  negative  ;  therefore,  in  any  equation  reduced  to  the 
above  form,  if  r •<[ .,,.     .,  there  is  a  real  small  positive. root,  <C '-'''■ 

EXAMPLE. 

r»_|_l&r3— 2l2-2— 12x+l  =  0 

has  a  real  root  between  0  and  -. 

u 

287.  To  find  an  inferior  limit  of  the  positive  roots,  we  must  transform  tho 
equation  into  one  whose  roots  are  the  reciprocals  of  the  roots  of  the  former ; 
and  tho  reciprocal  of  tho  superior  limit  of  the  roots  of  tlie"  transformed  equa- 
tion, found  by  the  preceding  methods,  will  be  the  quantity  required. 

Hence,  if  j>r  denote  tho  greatest  coefficient  of  a  contrary  sign  to  the  last 

m 

term,  p„,  an  inferior  limit  of  the  positive  roots  is ; .     For  the  transformed 

equation  will  be  (Art.  271) 

Pn-l  Pi  1 

yn+ — ya~l-\ \—yr+  ...+—= o, 

Vr  Vi 

of  which  —  is  tho  greatest  negative  coefficient ;  therefore       4- 1  is  a  supenoi 

Pa 
limit  of  its  roots;  and,  consequently,  — — —  an  interior  limit  of  the  positive  root* 

o   the  proposed  equation. 


A13W  TON'S  METHOD  OF  FINDING  THE  LIMITS  OF  THE  ROOTS.    365 

I 

EXAMPLE. 

jJ_42x2+441x— 49=0. 

49  1 

Here  pQ=49,j?r=  141,  .-.    f   ,    .  .    ,  or  —  is  an  inferior  limit  of  the  positive 

roots.     By  putting  x=-,  we  may  discover  a  limit  Cjoser  to  tho  roots ;  for  the 

*j 

transformed  equation  is 

6         1  6/         1\ 

y*-9y*+-y--=0,oi-if(y-9)  +  -[y--)=0, 

which  evidently  has  9  for  the  superior  limit  of  its  positive  roots,  and,  theie 
fore,  the  proposed  has  -  for  its  inferior  limit. 

288.  To  find  superior  and  inferior  limits  of  the  negative  roots,  we  must 
transform  the  equation  into  one  whose  roots  are  those  of  the  former  with  con- 
trary signs  (Art.  247) ;  and  if  a,  /3  bo  limits,  found  as  above,  of  the  positive 
roots  of  this  equation,  then  — a  and  — /?  will  be  limits  of  the  negative  roots  of 
the  proposed  equation. 

EXAMPLE. 

a«_  7z+7=Q; 

putting  .r= — y,  we  get  y" — 7y — 7=0,  of  which  1+  y/7  or  4  is  a  superior 

limit. 

1  113 

Also,  putting  y=-,  we  get  z»+z2— -=0,  or  z3— — +z3— —  =  0,  of  which 

-  is  a  superior  limit ;  therefore  the  negative  root  of  the  proposed  lies  between 
o 

—4  and  —3. 

newton's  method  of  finding  limits  of  the  roots. 

289.  The  limits,  however,  deduced  by  any  of  the  preceding  methods  sel- 
dom approach  very  near  to  the  roots ;  the  tentative  method,  depending  upon 
the  following  proposition,  will  furnish  us  with  limits  which  lie  much  nearer  to 
them. 

Every  number  which,  written  for  x,  makes/(x)  and  all  its  derived  functions 
positive,  is  a  superior  limit  of  the  positive  roots. 

For,  if  we  diminish  the  roots  a,  b,  c,  &c,  of/(x)  =  0  by  h,  that  is  (Art.  251V 
substitute  y-\-h  for  x,  the  result  isf(y-\-h)=0,  or 

m+fv>)i  +/'/(/o1^7+  •••  +/n-'(/o^+2/n=o- 

Now,  if  we  give  such  a  value  to  h  that  all  the  coefficients  of  this  equation 
are  positive,  then  every  value  of  y  is  negative ;  that  is,  all  the  quantities,  a — h, 
ft — ht  c—h,  &c,  are  negative,  and  therefore  h  is  greater  than  the  greatest  of 
the  quantities  a,  b,  c,  &c,  or  is  a  superior  limit  of  the  roots  of  the  proposed 
equation.  Similarly,  h  will  be  an  inferior  limit  to  al  the  roots,  if  the  coefficients 
be  alternately  positive  and  negative. 

EXAMPLE. 

To  find  a  superior  limit  of  the  roots  of 

z3  —  oj?+7x— 1  =  0. 


3G6  ALGEBRA. 

The  transformed  equation,  putting  y-\-h  for  x,  is 

(h3—5h*+7h  —  l)  +  {3lr—U)h  +  7)ij+(6h-..10)^+y=Q; 

in  which,  if  3  be  put  for  h,  all  the  coefficients  are  positive;  therefore  3  is  a  su- 
perior  limit  of  the  positive  roots. 

Observation. — This  method  of  finding  a  superior  limit  of  the  roots  by  de- 
termining by  trial  what  value  of  x  will  make/(.r)  and  all  its  derived  functions 
positive,  was  proposed  by  Newton. 

waring's  or  lagrange's  method  of  separating  the  roots. 

290.  If  a  series  of  quantities  be  substituted  for  x  in/(.r),  then  between  every 
two  which  give  results  with  different  signs  au  odd  number  of  roots  of/"(.r)  =  0 
is  situated;  and  between  eveiy  two  which  give  results  with  the  same  sign  an 
even  number  is  situated,  or  noue  at  all;  but  we  can  not  assure  ourselves  that 
in  the  former  case  the  number  does  not  exceed  unity,  or  that  in  the  latter  it 
is  zero,  ami  that,  consequently,  the  number  and  situation  of  all  the  real  roots 
is  ascertained,  unless  the  difference  between  the  quantities  successively  sub- 
stituted be  less  than  the  least  difference  between  the  roots  of  the  proposed 
equation ;  since,  if  it  were  greater,  it  is  evident  that  more  than  one  root  might 
be  intercepted  by  two  of  the  quantities  giving  results  with  different  signs,  and 
that  two  roots  instead  of  none  might  be  intercepted  by  two  of  the  quantities 
giving  results  with  the  same  sign,  and  in  both  cases  roots  would  pass  undis- 
covered. We  must,  therefore,  first  find  a  limit  less  than  the  least  difference 
of  the  roots ;  this  may  be  done  by  transforming  the  equation  into  one  whose 
roots  are  the  squares  of  the  differences  of  the  roots  of  the  proposed  equation. 
Then,  if  we  find  a  limit  k  less  than  the  least  positive  root  of  the  transformed 
equation,  y7 k  will  be  less  than  the  least  difference  of  the  roots  of  the  proposed 
equation;  and  if  we  substitute  successively  for  x  the  numbers  s,  s — V k, 
s  —  2 -//>">  &c-  (s  being  a  superior  limit  of  the  roots  of  the  proposed),  till  we 
come  to  a  superior  limit  of  the  negative  roots,  we  are  sure  that  no  two  real 
roots  lying  between  the  numbers  substituted  have  escaped  -.  ami  that  every 
change  of  6igns  in  the  results  of  the  substitutions  indicates  only  one  real  root. 
Hence  the  number  of  real  roots  will  be  known  (for  it  will  exactly  equal  the 
number  of  changes),  as  well  as  the  interval  in  which  each  of  them  is  contained. 

Observation. — This  method  of  determining  the  number  and  situation  of 
the  real  roots  of  au  equation  was  first  proposed  by  Waring  :  ii  is,  however,  of 
no  practical  use  for  equations  of  a  degree  exceeding  the  fourth,  on  account  of 
the  great  labor  of  forming  the  equation  of  differences  for  equations  of  a  higher 
order. 

I  \  LHPLE. 

xi— 7x+7=0. 

The  numbers  1  and  2  give  each  a  positive  result,  but  yet  two  roots  lie  be- 
tween them.  The  equation  whose  roots  are  the  squares  of  the  differences  is 
(Art. 279)  i/3 — 42?/a-f-44l2/ — 49  =  0.  an  inferior  limit  of  the  positive  roots  of 

which  is  -  (Art.  287);  therefore,  -  is  loss  than  the  least   difference  of  the 
9  o 

5   4 
roots  of  Is  —  7.r-f-7  =  (>,  ami,  substituting  2,  -,  .,,  the  results  are  -|-,  — •  -+■ 


METHOD  OF  DIVISORS.  3C7 

5  5       i  4 

hence   one  value  of  x  lies  between  2  and  -,  and  one  between  -  and  -  •,  and, 

O  O  u 

similarly,  we  find  the  negative  root,  which  necessarily  exists,  to  lie  between  — 3 
and  — 3x. 

METHOD   OF  DIVISORS. 

291.  The  commensurable  roots  of/(x)  =  0,  which  are  necessarily  whole 
numbers,  may  bo  always  found  by  the  following  process,  called  the  method  of 
divisors,  proposed  by  Newton. 

Suppose  a  to  be  an  integral  root;  then,  substituting  a  for  .r,  and  reversing 
the  order  of  th,e  terms,  we  have 

Pn+i7n-ia+pn-2a3+  . . .  -f^a"-1 +0"= 0  ; 

•••  ^+P"->  +JW*  -\ —  +^i«n-2+«n_1  =  o- 

Hence,  —  is  an  integer  which  we  may  denote  by  qi ;  substituting  and  di- 
viding again  by  a,  we  get 

?'+aPn~1+Pn-2+  •  •  •  +lha"-3+a»-*  =  0. 

Similarly,  "~1  is  an  integer  =q2  suppose  ;  and  proceeding  in  this  man- 

ner, we  shall  at  last  arrive  at 

a        ' 

Hence,  that  a  may  be  a  root  of  the  equation,  the  last  term,  pa,  must  be  di- 
visible by  it,  so  must  the  sum  of  the  quotient  and  next  coefficient,  qx-\-pD-i ; 
and  continuing  the  uniform  operation,  the  sum  of  each  coefficient  and  the  pre- 
ceding quotient  must  be  divisible  by  a,  the  final  result  being  always  —1. 

If,  therefore,  we  take  the  quotients  of  the  division  of  the  last  term  by  each 
of  the  divisors  of  the  last  term  which  are  comprised  within  the  limits  of  the 
roots,  and  add  these  quotients  to  the  coefficient  of  the  last  term  but  one ;  di- 
vide these  sums,  some  of  which  may  be  equal  to  zero,  by  the  respective 
divisors,  add  the  new  quotients  which  are  integers  or  zero  (neglecting  the 
others)  to  the  next  coefficient  and  divide  by  the  respective  divisors,  and  so  on 
through  all  the  coefficients  (dropping  every  divisor  as  soon  as  it  gives  a  frac- 
tional quotient),  those  divisors  of  the  last  term  which  give  —1  for  a  final  re- 
sult are  the  integral  roots  of  the  equation ;  and  we  shall  thus  obtain  all  the  in- 
tegral roots,  unless  the  equation  have  equal  roots,  the  test  of  which  will  be  that 
some  of  the  roots  already  found  satisfy /'(x)  =  0,  and  the  number  of  times  that 
any  one  is  repeated  will  be  expressed  by  the  degree  of  derivation  of  the  first 
of  the  derived  functions  which  that  root  does  not  reduce  to  zero,  when  written 
in  it  for  x  (Art.  253).  It  is  best  to  ascertain  by  direct  substitution  whether 
-\-l  and  — 1  are  roots,  and  so  to  exclude  them  from  the  divisors  to  be  tried. 

EXAMPLE  I. 

a«4-3a*— 8a:+10=0. 

Q 

Here  the  roots  lie  between  -+1  and  —11  (Arts.  285,  288),  and  the  divi- 
sors of  the  last  terra  are  ±  {2,  5,  10 }, 


3(J8  ALGEBRA. 

.:  a  =  2  —   2  —  5  —10 

g1==  5  —  5  —   2  —   1 

?l  +  (_8)=  —3  —13  —10  —  9 

']:=  2 

?s=  —  !• 

Therefore  — 5,  being  the  only  one  of  the  divisors  which  leads  to  a  last  quo- 
tient —  1,  is  the  only  commensurable  root,  and  it  is  not  repeated,  since  it  does 
not  satisfy  the  equation /'(.r)  =  3.r2-r-  6x — 8=0. 

EXAMPLE  II. 

zs_5xi+:r>-f  i6x2— 20x+16  =  0. 
Here  limits  of  the  roots  are  6  and  —4  ;  and  the  commensurable  roots  are 

EXAMPLE   III. 

a-i_|_5x-3_ox2_6x-r-20=0;  x=  —  2,  or  —5. 

292.  The  number  of  divisors  to  be  tried  may  be  lessened  by  observing,  that 

if  the  roots  of  /(.r)  =  0  were  diminished  by  any  whole  number,  m,  the  last 

term  of  the  transformed  equation,  f{y-\-m)  =  0,  would  bef{m) ;  if,  therefore, 

a  were  an  integral  value  of  x,  a—m  would  bo  an  integral  value  of  y,  and  would 

be,  therefore,  a  divisor  of  f(m).     Hence,  any  divisor,  a,  of  the  last  term  of 

f(m) 
f(x)  is  to  be  rejected  which  does  not  satisfy  the  condition  -— —  =  an  integer, 

when  for  m  any  integer,  such  as  ±1,  ±10,  &c,  is  substituted. 

EXAMPLE  I. 

r5— 5.r2  — 181+72  =  0. 
Changing  the  signs  of  the  alternate  terms,  we  have 

ar»+5a«— 18r— 72=0,  or  a*— 78+6r(x— r-)  =0-, 

therefore-  the  roots  lie  between  19  and  — 5. 

But  /(l)=50,/(-l)=S4,/(-3)=54; 

and  the  only  admissible  divisors  of  72,  which,  when  diminished  by  1,  divide 
50,  are 

G,  3,  2,  -4  ; 

also,  all  these  divisors,  when  increased  by  1,  divide  84;  but  only  6,  5,-4 
when  increased  by  3,  divide  54  ; 

.-.  G,  3,  —4, 

are  the  only  divisors  which  need  to  be  tried  ;  and  they  will  all  bo  found  to  be 
roots. 

EXAMPLE  II. 

:j3_  6i»+169x— (42)8=0.     .r=9. 

293.  If  a  proposed  equation  have  fractional  coefficients,  or  if  its  first  term 
be  affected  with  a  coefficient,  since  (275,  Cor.  2)  it  can  be  transformed  into  an- 
other equation  with  first  term  unity  and  every  coefficient  a  whole  number 
this  method  will  onablo  us  to  find  the  commensurable  roots  of  every  equation 
undor  a  rational  form.  If  the  coefficients  be  whole  numbers  and  the  first  term 
be^v",  and  wo  only  wish  to  find  the  roots  which  are  integers,  no  transfcrmtr 


NEWTON'S  METHOD  OF  APPROXIMATIC  N.  369 

turn  will  bo  necessary,  only  every  divisor  of  the  last  term  which  is  a  root  will 
lead  to  a  result  — p0  instead  of  —1. 

EXAMPLE. 

6x*— 253?-\-26x-+4x— 8  =  0. 

It  is  the  same  as 

{x— 2)2(3z— 2)(&c+ 1) =0. 

newton's  method  of  approximation. 

294.  When  we  know  an  approximate  value  of  a  root,  we  may  easily  obtain 
other  values  of  it,  more  and  more  exact,  by  a  method  invented  by  Newton, 
which  rapidly  attains  its  object.  We  shall  give  this  method,  first  in  the  form 
in  which  it  was  proposed  by  its  author,  and  afterward  with  the  conditions 
which  Fourier  has  shown  to  be  necessary  for  its  complete  success. 

Let/(z)  =  0  be  an  equation  having  a  root  c  between  a  and  b,  the  difference 
of  theso  limits,  b  — a,  being  a  small  fraction  whose  square  may  be  neglected  in 
the  process  of  approximation. 

Let  Ci,  a  quantity  between  a  and  b,  be  assumed  as  the  first  approximation 
to  c,  then  c=cl-\-h,  where  h  is  very  small ; 

.-.  7(^4-70=0, 

01 

f(ci)+f'(ci)h+f"(cl)^+  . . .  +h»=0. 

Now,  since  h  is  very  small,  /j2,  /t3,  &c,  are  very  small  compared  with  h ; 
also,  none  of  the  quantities/"(ci),/'"(ci),  &c,  can  become  very  great,  since  they 
result  from  substituting  a  finite  value  in  integral  functions  of  x;  therefore,  pro- 
vided f'(ci)  be  not  very  small  (that  is,  provided  /'(.r)=0  have  no  root  nearly 
equal  to  Ci  or  to  c,  and,  consequently,  f{x)  =  0  no  other  root  nearly  equal  to  c 
besides  the  one  we  are  approximating  to),  all  the  terms  in  the  series  after  the 
first  two  may  bo  neglected  in  comparison  with  them  ;  and  we  have,  to  deter- 
mine h,  the  resulting  approximate  value  of  h,  the  equation 

/(ci)+Vte)=<>; 

•  i~  net)-  «/(*)>—/ 

and  the  second  approximation  is 

_L.7  U{X)1 

Similarly,  starting  from  ca  instead  of  eu  the  third  approximate  value  will  be 

$  A*)  I 

C3-C-   I  fi{x)    $  _; 

and  so  on;  and  if  we  can  be  certain  that  each  new  value  is  nearer  to  the  truth 
than  the  preceding,  there  is  no  limit  to  the  accuracy  which  may  be  obtained. 

EXAMPLE  I. 

x3— 2x— 5=0. 
Here  one  root  lies  between  2  and  3,  and  the  equation  can  have  only  on» 


*  This  notation  signifies,  that  after  the  division  indicated  is  performed,  the  particular 
vmlue  ci,  is  substituted  for  x. 

A  A 


370  ALGEBBA. 


positive  root;  also,  upon  narrowing  the  limits,  we  fiud  that  i=2  gives  a  nega- 
tive, and  ar=2-2  a  positive  result;  therefore,  2-1  differs  from  the  root  bv  a 
quantity  less  than  0-1,  and  wo  may  assume  ct=2'l.     Hence 

(x3— 2j— 5\ 


0-061 
11-23' 


or 


Similarly, 


c3=2-l— 0-0054=2-0940. 
c3= 2-  094  55 14  9. 


EXAMPLE  II. 

Xs— 7x— 7  =  0. 

There  is  only  one  positive  root  lying  between  3  and  3-1,  and  it  equal* 
3-016917339. 

Observation. — To  guard  against  over  correction,  that  is,  against  appl 
such  a  correction  to  an  approximate  value  as  shall  make  the  new  value  differ 
more  from  the  root  by  excess  than  the  original  approximate  value  did  by  de- 
fect, or  vice  versa,  we  must  be  certain  that  each  new  value  is  nearer  to  rue 
truth  than  the  preceding;  this  gives  rise  to  the  following  conditions,  first  no 
ticed  by  Fourier. 

295.  For  the  complete  success  of  Newton's  method  of  approximation,  the 
following  conditions  are  necessary. 

1.  The  limits  between  which  the  required  root  is  known  to  lie  must  be  so 
close  that  no  other  root  of/(x)=0,  and  no  root  of/  (i)=0,  or/"(.r)  =  0,  lies 
between  them. 

2.  The  approximation  must  be  begun  and  continued  from  that  limit  which 
makes  f{x)  and/"(r)  have  the  same  sign. 

Let  c  be  a  root  of/(.r)=0  which  lies  between  a  and  6,  a<6,  cx  the  first  ap- 
proximate value,  and  h  the  whole  correction,  so  that  c=Ci -f-/t ;  then 

/(cl+A)=0,or/(c1)+V''(i)=0. 

A  being  some  quantity  between  c,  and  c  (Art.  239,  Note). 

Therefore,  supposing  A=c,,  which  amounts  to  neglecting  all  powers  of  h 
above  the  first,  and  requires  that/(.r)  =  0  have  no  root  besides  c  in  that  interval, 
and  calling  the  resulting  approximate  value  of  h,  hu  we  have 

/(c,)+V(ci)=0. 

Now  the  true  value  is  c=cl-\-h  ; 

The  first  approximate  value  is  ct  with  error  //  ; 

The  second  approximate  value  is  ci=Ci-^-hl  with  error  h—hu  which  (ueg 
lecting  signs)  must  be  less  than  h, 

i.  e.,  h* — (h — hi)9  must  be  positive,  or  2/j/;, — hl'2  =  -{-, 

or^-'=+'or7W-i=+: 

which  condition  (since  A  is  an  indeterminate  quantity  between  c,  and  r,  or  be- 
tween a  and  b)  can  not  in  all  cases  be  secured  unless /'(.r)  be  incapable  of 
changing  its  sign  between  a  and  b,  i.  C,  unless /'(x)r=0  have  no  root  between 
a  and  b. 

Moreover,  we  must  have  j77^\>^  or  >1,  the  latter  insuring  the  former. 

Now,  if/"(x)  preserve  an  invariable  aign  between  a  and  />,  i.  c,  \(f"(r)=Q 


NEWTON'S  METHOD  OF  APPROXIMATION*.  371 

have  no  root  in  that  interval,  then  f'(r)  will  increase  or  diminish  contirually 
from  a  to  b  ;  therefore  Ci  must  be  taken  equal  to  that  limit  which  gives  /'(<) 
its  greatest  numerical  value  without  regard  to  sign. 

First,  let/'(.r),/"(.r),  have  the  same  sign  from  a  to  b  ;  thcn/'(.r)  increases 
continually  in  that  interval;  therefore  we  must  have  Cj=6,  or  we  must  begin 
from  the  greater  limit.  But/(6)  has  the  same  sign  as  f(c-{-h)=f(c)-\-hf'{c) 
=hf'(c),  or  as /'(c) ;  therefore  we  must  have  cx  equal  to  that  limit  which  makes 
f(x)  andf'(x)  have  the  same  sign. 

Secondly,  ]et  f'(x),  f"(x),  have  contrary  signs  from  a  to  b  ;  then/'(.r)  di- 
minishes continually  in  that  interval ;  therefore  we  must  have  Ci=a,  or  we 
must  begin  from  the  lesser  limit.  But  f(a)  has  the  same  sign  as  f(c — h) 
=f(c) — hf'(c)  =  — hf'(c),  or  as  —/'(c);  therefore,  in  this  case,  equally  as  in 
the  former,  we  must  have  cx  equal  to  that  limit  which  makes  f(x)  and/"(.r) 
have  the  same  sign. 

Theso  conditions  being  fulfilled,  we  have 

''<"'>    .l=+,or*  -■ 


c—c2 


or 


C2  — Ci 

therefore  <■,  lies  between  c  and  Ci ;  hence,  the  new  limit,  c2  fulfills  the  requi- 
site conditions,  and  wo  may  with  certainty  from  it  continue  the  approxima- 
tion. 

296.  To  estimate  the  rapidity  of  the  approximation,  we  have 

error  in     first    approximate  value  cu  ssk, 
error  in  second  approximate  value  c2,  =h — h{ ; 
But  f(ci)+hf>(Cl)  +  lh*f"(n)=0, 

/(ci)+/->/'(<->)=0; 

.-.  (h-ih)f'{Cl)+yii*fy)=o, 

"     /'(ci) 
Let  the  greatest  value  which  f"(x)  can  assume  between  a  and  b  (whicn 
will  be  either/"(a)  or f  "(b),  if /'"(.r)  =  0  have  no  root  in  the  interval)  be  di- 
vided by  the  least  value  of  2/'(.r)  in  that  interval  which  will  be  either  2f'(a)  or 
2/'(6),  and  let  the  quotient  be  denoted  by  C  ;  then,  neglecting  signs, 

hence,  if  the  first  error  h  in  ^  be  a  smad  decimal,  the  error  h — hi  with  which 
c2  is  affected  (since  C  will  not,  except  in  particular  cases,  be  very  large)  will 
be  very  small  compared  with  h  ;  and  if  tho  quantity  C  be  less  than  unity,  the 
number  of  exact  decimals  in  the  result  will  be  doubled  by  each  successive 
operation.  The  quantity  C,  when  thus  computed  for  a  given  interval,  pre- 
serves the  same  value  throughout  the  operations  which  it  may  be  necessary  to 
make  in  order  to  approximate  to  the  value  of  the  root  lying  in  that  interval ; 
and  as  we  thus  know  a  limit  to  the  difference  between  the  approximate  value 
already  found  and  the  true  value,  we  may  always  avoid  calculating  decimals 
which  are  inexact,  and  only  obtain  those  which  are  necessarily  correct. 

EXAMPLE. 

6X3— 141x+263=0. 
This  equation  has  two  positive  roots,  one  between  2-7  and  2-8,  and  the 


M2  ALGEBRA. 


other  between  2-8  and  2-9.     Now  f'(x)=18xa — 1-11  =  0  has  a  root 


-£ 


=  2-798,  betwoen  2-7  and  2-8,  therefore  these  limits  are  not  sufficiently  close  ; 
but  this  root  is  greater  than  2-79  ;  also,  2-7  and  2'79,  substituted  in/(x),  give 
results  with  different  signs;  and  2-7,  substituted  in  f(x)  and  f"(x),  gives  re- 
sults with  the  samo  sign  ;  therefore,  c,=2-7. 

With  regard  to  the  other  interval,  2-8,  2-9,/'(.r)  =  0./"(.r)=0  have  no  roots 
between  these  limits,  and  2-9  makes /(x)  and  f"(x)  have  the  same  sign; 
therefore,  c,:=2-9;  and  starting  from  these  values,  we  are  certain  in  each 
case  to  get  a  value  nearer  to  the  truth. 

f"{x) 

Again,  the  greatest  value  which  ^,  can  assume  in  the  interval  2-7, 

2-79,  is  nearly  equal  to  10 ;  hence,  if  hi,  h2,  be  consecutive  errors,  we  have 

The  samo  formula  will  be  found  to  be  time  for  consecutive  errors  in  the  in 
terva]  2  8,  2-9. 

IiAGRANGE's  METHOD   OF  APPROXIMATION  BY   CONTINUED   FRACTIONS. 

297.  To  approximate  to  the  roots  of  an  equation  by  the  method  of  continued 
fractions. 

Let  the  equation  f(x)  =  0  have  only  one  root  between  the  integers  a  and 

a-\-l  ;*  then,  writing  a-\--  for  x,  the  first  transformed  equation  will  be 

/(«)+/'(^+/»n^.+  -+^=o  (i); 

and,  since  only  one  value  of-  lies  between  0  and  1,  y  has  only  one  value  greater 

than  1 ;  if,  therefore,  we  substitute  successively  2,  3,  4,  &c,  for  y,  stopping 
at  the  first  which  gives  a  positive  result,  the  integer  preceding  that  is  the  in 

tegral  part  of  the  value  of  y.     Let  this  be  b,  and  in  (1)  write  b-\--  for  y  ;  then 

the  second  transformed  equation  will  have  only  one  root  greater  than  unity, 
the  integral  part  of  which,  as  before,  will  be  the  whole  number  next  less  than 
tho  one  in  the  series  2,  3,  4,  cVc,  which  first  gives  a  positive  result  when 
written  for  z  ;  let  this  bo  c,  and  in  the  second  transformed  equation  write 

c-|-~  for  z,  then  the  third  transformed  equation  will  have  only  one  root  greater 

than  unity,  the  integral  part  of  which  may  bo  found  as  before,  and  so  on. 
We  thus  obtain  successively  the  terms  of  a  continued  fraction 

c+d'  &c- 
which  expresses  the  required  valuo  of  x.     The  method  of  reducmg  such  a 
tion,  callod  a  continued  fraction,  will  be  hereafter  given. 

*  Tho  roots  of  the  equation  may  be  made  to  differ  by  at  least  unity,  if  wo  fnJ  by  meani 
of  the  equation  of  tho  squares  of  die  differences  the  1  it  limit  to  tho  differences  of  tho 
roots  of  the  proposed  equation,  and  then  find  n  brand  quation  whose  roots  shall  bo 

that  multiple  <>f  those  of  the  proposed,  which  is  expressed  by  the  denominator  of  tho  least 
limit  of  tho  differences. 


LAGRANGE'S  METHOD  OP  APPROXIMATION  373 

If  any  of  the  numbers  b,  c,  d,  &c,  is  an  exact  root  of  the  101  responding 
transformed  equation,  the  process  terminates,  and  we  find  the  exact  value  of  a 
Also,  if  one  of  the  transformed  equations  be  identical  with  a  preceding  one, 
the  continued  fraction  expressing  the  root  is  periodical;  for,  after  that,  the 
same  quotients  will  recur  in  the  same  order ;  in  this  case  a  finite  value,  in  the 
form  of  a  surd,  may  be  obtained  for  the  root  (see  Continued  Fractions)  by  solv- 
ing a  quadratic  whose  coefficients  are  rational,  both  of  whose  roots  will  be  roots 
of  the  proposed,  sinco  the  coefficients  of  the  latter  are  supposed  rational ;  con- 
sequently, the  first  member  of  this  quadratic  will  be  a  factor  of  the  first  mem- 
ber of  the  proposed  equation,  which  may,  therefore,  be  depressed  two  di- 
mensions. 

EXAMPLE. 

To  find  the  positive  root  of  xz — 2.r — 5=0  under  the  form  of  a  continueu 
fraction. 

Comparing  this  with  x3 — qx-\-r=0,  we  find  that 

r2      qs      25      8 

-——=———  is  a  positive  quantity  ; 

therefore  (Art.  258)  the  equation  lias  two  impossible  roots ;  and  since  its  last 
term  is  negative,  its  third  root  is  positive.     Substituting  2  and  ?,  the  results  are 

— 1  and  +16;  therefore  the  root  lies  between  2  and  3.     Assume  x=24--i 

y 

and  the  transformed  equation  is 

2/3_107/"-_67/-l=0, 

in  which  10  and  11  being  substituted,  give  — 61,  +54.     Assume  i/  — 10+-i 

and  we  obtain 

Gl:3— 94--— 20z  — 1  =  0, 
whose  root  lies  between  1  and  2.     Proceeding  in  this  manner,  we  find 

1111 

•r=2+Io-+i+T+2--- 

the  value  of  the  root  in  a  continued  fraction  ;  the  method  of  reducing  which 
to  a  common  fraction  will  be  hereafter  given. 

This  method  may  be  combined  with  Sturm's  theorem. 

Here  finishes  our  recapitulation  of  the  older  methods.  What  follows  be- 
longs to  the  present  more  improved  state  of  algebraic  science.* 


*  We  shall  here  point  out  a  method  of  finding;  the  equal  roots  of  an  equation,  which 
avoids  the  laborious  process  of  seeking  the  common  divisor,  and  which  may  be  employed 
when  any  other  than  Sturm's  process  for  discovering  the  roots  of  an  equation  is  used. 

1.  If  an  equation  whose  coefficients  are  commensurable  have  a  pair  of  equal  roots  and  no 
greater  number,  these  roots  must  be  commensurable  ;  for  the  common  measure  of  the  first 
member  of  this  equation,  and  the  function  derived  from  it,  will  be  a  binomial  expression  of 
the  first  degree  with  finite  coefficients,  and  which,  when  equated  to  zero,  will  furnish  one 
of  the  equal  roots  ;  these  roots,  therefore,  must  be  commensurable,  that  is,  either  integers 
or  fractious. 

2.  If  the  leading  coefficient  in  the  supposed  equation  be  unity,  and  the  others  integral, 
the  equal  roots  mist  be  integral,  because  no  fractional  root  can  exist  under  these  condi- 
tions (Art.  246).  , 

3.  If  an  equation  with  commensurable  coeffici  ;nts  have  three  equal  roots,  and  no  more, 
these  also  must  be  commensurable ;  for,  in  this  case,  the  common  measure  will  be  of  the 
second  degree,  and,  when  equated  to  zero,  will  give  two  of  the  equal  rorts  :  these  mots,  !  s 
just  remarked,  must  be  coninici>surable  ;  hence  all  the  three  roots  must  be  commensurable 


374  ALGEBRA. 

BINOMIAL  EQUATIONS. 
298.  Binomial  equations  are  those  which  can  be  reduced  to  the  t  rra 

xm=A  ori"1  —  A  =  0 (1) 

A  being  any  known  quantity  whatsoever. 


An  I,  as  before,  if  the  leading  coefficient  be  unity,  and  the  others  integral,  the  equal  roott 
will  be  integral. 

4.  By  the  same  reasoning,  if  an  equation  with  commensurable  coefficients  have  m  equal 
roots,  and  no  other  groups  of  equal  roots,  these  m  roots  must  be  commensurable  ;  and  they 
will  be  integral  if  the  leading  coefficient  be  unity  and  the  other  oo<  integers. 

5.  When  the  leading  coefficient  is  unity,  and  the  other  coefficients  whole  numbers,  and 
/«  equal  integral  roots  enter,  we  may  infer,  from  the  formation  of  the  coefficients  (245),  that 

absolute  number,  and  the  coefficient  of  the  immediately  preceding  term,  that  is,  the 
licient  of  x,  will  admit  of  a  common  measure  involving  m — 1  of  these  roots ;  that  the 
coefficients  of  x  and  x"  will  have  a  common  measure  involving  m — 2  of  them ;  and  so  od 
till  we  come  to  the  coefficients  of  a**-*  and  a"*-',  which  will  have  a  common  measure  in- 
volving the  multiple  root  once.  For,  if  the  depressed  equation  containing  only  the  unequal 
roots  be  considered,  it  will  involve  none  but  integral  coefficients,  since  its  last  term  is  form- 
ed from  the  penult  coefficient  of  the  proposed  divided  by  one  root ;  so  that  if  the  equal  roots 
be  now  introduced,  they  can  combine  with  none  but  integral  factors.  Hence,  if  the  root  occur 
twice,  it  will  be  found  among  the  integral  factors  of  the  common  measure  of  the  coefficients 
An  (the  final  coefficient)  and  A„-i  ;  if  it  occur  three  times,  it  will  be  found  among  the  fac- 
tors of  the  common  measure  of  An,  An-i,  and  An-e,  and  so  on.  And,  therefore,  by  trying 
several  factors  of  the  common  measure  in  question,  by  actually  substituting  them  for  x  in  the 
proposed  equation,  when  from  any  circumstance  multiple  roots  are  suspected  to  exist,  we 
may  remove  all  doubt  on  the  subject.  In  analyzing  an  equation,  the  doubts  that  may  arise 
as  to  the  entrance  of  equal  roots  arc  confined  to  certain  defiuite  intervals,  or  within  deter- 
minate numerical  limits  ;  so  that,  of  the  factors  adverted  to  above,  only  those  falling  within 
these  limits  need  be  regarded. 

And  further,  if  the  repeated  root  occur  but  twice,  the  square  of  it  must  be  a  factor  of  a0 
or  An;  if  it  occur  three  times,  the  cube  of  it  must  be  a  factor  of  An,  and  the  square  of  it  a 
factor  of  An-i ;  if  it  occur  four  times,  the  fourth  power  of  it  must  be  a  factor  of  An,  the  cube 
of  it  a  factor  of  A„_i,  and  the  square  of  it  a  factor  of  An-:,  and  so  on.  And  thus,  of  the 
factors  of  A„  to  be  tested,  those  only  need  be  used  whose  powers  also  are  factors,  entering, 
as  here  described,  according  to  the  multiplicity  of  the  roots. 

6.  These  inferences  may  be  easily  generalized :  they  apply,  whatever  be  the  integral 
value  of  the  leading  coefficient,  and  whether  the  repeated  root  be  integral  or  fractional. 

Thus,  let  the  repeated  root  be  x=j,  a  and  b  having  no  common  factor ;  then,  if  the  root  en- 
ter m  times,  the  original  polynomial  will  be  divisible  by  (bx—a)m,  giving  a  quotient  in- 
volving the  n  roots,  and  into  which  none  but  intei  ral  coefficients  enter  (953).    Let 
us  now  return  to  the  original  polynomial  by  multiplying  this  quotient  by  bx—a  m  tin 
the  first  multiplication  by  bx—a  will  evidently  uive  a  product,  into  the  lirst  term  of  which 
b  must  enter  as  a  factor,  ami  into  tho  last  of  which  a  must  enter  ;  the  next  multiplication 
must,  therefore,  give  a  product,  into  the  Brat  term  of  which  b-  must  enter,  into  the  second 
b,  into  the  last  ./-,  and  into  the  last  but  one  a ;  the  third  multiplication,  therefore,  must 
give  a  product  whose  first  three  tonus  involvi  &  last  tbn 
a<  re              these  last  in  reverse  order,  and  so  on.    11.  nee  the  coefficients  Ai,  A..  A  .  Ac, 
will  be  divisible  by  bm,  bm~l,  ba    ',  fco.,  respectively,  down  to  b;  and  the  ooefflcii  nta  A  . 
An-i,  An  a.  Ac,  by  am,  a1"-',  a™-'-',  Ac.,  down  to  a.     In  other  words,  the  coefficients,  I 
in  order,  reckoning  from  the  beginning,  will  be  divisible  by  the  corresponding  decreasing 
powers  of  the  denominator  of  the  repeated  root;  and  the  coefficients,  reckoning  from  the 
will  be  divisible  by  the  like  powers  of  the              tor. 

7.  The  inferences* still  have  place1,  whatever  be  tin  of  the  multiple  factor  • 
;„  ■  the  proposed  polj  normal,  so  Ion  ■  as  this  factor,  as  weB  as  tb   i  J  porynonrid,  I 
none  bat  integral  coefficients.    This  is  plain,  fro,,,  the  reasoning  in  the  pr< 

which  remains  tl.e  :  ame.  BS  respects  the  entrance  of  the  factors  b,  fl,  Whether  the 
multiplier  bo  bx—a  or  b.i  ,n-f-  •  •  •  •  +"• 


BINOMIAL  EQ.UATION1-.  375 

Wo  perceive  immediately  that  the  m  roots  of  this  equation  are  different 
from  one  another ;  for  the  first  member  xm — A  has  no  common  factor  with  its 
derived  function  ma:™-1,  and  hence  the  proposed  equation  (Art.  253,  Schol.) 
can  not  have  equal  roots.  The  roots,  if  we  raise  them  to  the  power  m,  ought 
each  to  produce  A,  since  they  are  tho  same  as  tho  values  embraced  in  the  ex- 
pression x=  t/A.  We  know,  then,  that  this  radical  has  m  different  values  ; 
but  we  shall  recur  to  this  subject  again,  and  more  at  length. 

299.  When  m  is  any  composite  number,  tho  solution  of  equation  (1)  re- 
duces itself  to  the  solution  of  several  binomial  equations,  the  degrees  of  which 
are  the  factors  of  m. 

Suppose  m=2)qr,  instead  of  the  equation  2rPV=0,  we  can  take  the  equations 

xp=x',  z"i=.r",  x'"=.A, 

in  which  x',  x"  are  new  unknowns. 

It  is  evident  that,  after  we  have  solved  the  equation  x'"=A,  the  preceding 
equation  .r"'=.x"  will  make  known  the  values  of  x',  and  that  then  the  equa- 
tion x?=x'  will  give  all  tho  roots  of  the  proposed  equation.  This  agrees  with 
the  formula  demonstrated  in  the  theory  of  radicals  (Art.  63),  viz., 


\/\A^ 


'  VA=TA. 

300.  Designate  by  a  a  quantity  whose  mth  power  is  A,  and  take  x=ay. 
The  equation  zm  =  A  becomes  amym=.am  ;  dividing  by  am, 

hence  y  =  ™/l5  and,  consequently,  x=a,yi. 

We  conclude,  therefore,  that  the  roots  of  the  equation  xm=A  can  be  ob 
tainod  by  multiplying  one  of  them  by  the  roots  of  the  equation  i/m=l  ;  or   in 
general,  that  the  different  m,h  roots  of  a  quantity  can  be  obtained  by  multiply- 
ing on©  of  them  by  the  m"1  roots  of  unity. 

301.  Let  us  consider  more  particularly  the  case  in  which  A  is  a  real  quan- 
tity ;  and,  to  distinguish  the  hypothesis  of  A  being  positive  or  negative,  writo 
the  binomial  equation  in  this  form  : 

2m=±A (2) 

These  conclusions  will  greatly  simplify  the  research  after  equal  roots,  and  will  either 
enable  us  wholly  to  dispense  with  the  laborious  process  for  the  common  measure,  or  will 
at  least,  render  the  more  tedious  steps  of  it  unnecessary 

EXAMPLES. 
2.1^—12^+19x2—6x4-9=0 (1) 

a;7+5i-fi4-6x5— 6.r*— 15a-3— 3x2-f  83+4=0    .  .  .  (2) 

The  first  of  these  can  have  no  fractional  repeated  roots,  because  the  leading  coefficient 
2  has  in  factor  a  perfect  power;  the  equal  roots,  if  any,  must,  therefore,  be  integral. 
Unity,  which  always  has  claims  to  be  tried,  does  not  succeed  ;  and  from  the  factors  of  9 
and  G,  it  is  plain  that  +3  and  — 3  arc  the  only  other  numbers  to  be  tested  ;  and  as  no 
higher  power  of  3  than  the  square  enters  9,  we  infer  that  more  than  two  equal  roots  can 
not  have  place  in  the  equation.  By  testing  3,  we  find  this  to  be  one  of  a  pair  of  equal 
roots.  Equal  quadratic  factors  could  not  possibly  enter  the  equation,  since,  as  the  first  co- 
efficient shows,  the  polynomial  is  not  a  complete  square.  In  the  second  of  the  above  equa- 
tions no  fractional  roots  can  enter.  Applying,  therefore,  +1  and  — 1,  we  discover  that 
-(-1  is  twice  a  root,  and  — 1  three  times.  The  remaining  equal  roots  — 2  and  — 2  are 
found  from  the  resulting  quadratic  obtained  by  suppressing  from  the  given  equation  the 
five  factors  of  the  first  degree. 


376  ALGKIJRA. 

Wo  can  determine,  at  least  by  approximation,  ft  positive  quantity  a  such 
that  wo  have  a'"=A.     Take,  again,  xssay,  equation  (2)  will  become 

This  is  tlio  equation  to  which  I  BhaU  confine  myself  exclusively. 

302.  'Tin-  following  remarks  may  be  made  with  regard  t<>  this  equation: 

1.  Winn  m  is  an  odd  number,  and  tin-  equation  is  ym  =  l  or  ?/m— 1=0,  it 
evidently  lias  the  root  <y=l  :  and   it    has  no  oilier  real  root,  for  every  other 
positive  value  of  y  will  give  y'n^>l  or  tfm<l,  and  a  negative  valuo  will  render 
ym  negative.     To  obtain  the  equation  on  which  the  ?« —  1  imaginary  roots  de 
pend,  we  shall  divide  ym  —  1  by  y — L,  and  thus  obtain  the  equation 

which  belongs  to  the  class  of  equations  called  reciprocal. 

2.  When  ///  is  an  odd  number,  and  the  equation  is  y'n= — 1,  it  has  evi 
dently  lor  a  root  y  = — 1.  By  a  reasoning  analogous  to  the  preceding, 
it  may  be  proved  that;  the  other  roots  arc  imaginary;  and  wo  obtain  the 
equation  on  which  they  depend  by  dividing  y'n-\-l=0  by  y-\-\.  13ut  to 
obtain  all  the  roots  of  tlio  equation  ym=  —  1,  it  is  well  to  remark  that  this 
equation  can  be  derived  from  ymss —  1  by  changing  y  into  — y.  It  will  suffice, 
then,  to  take  all  tlio  roots  of  i/'"  =  l  with  contrary  signs. 

3.  Supposo  m  is  an  ovon  number,  and  let  ?/i  =  2rt,  the  equation  y"n  —  1,  01 
y"n  — 1  =  0,  has  fur  its  roots  y  =  -\-l  ami  y=  —  1.  Tho  other  loots  are  imagin- 
ary, and  the  equation  which  contains  them  can  bo  obtained  by  dividing  '/  — 1 
=0  by  (?/ — l)(?/-f-l),  or  y"  —  1  ;  but  it  will  bo  well  to  observe  that  y"n — ) 
=  (?/" —  l)(.7/n+l)»  nnil  that,  consequently,  the  equation  if  — 1  =  0  can  bo  re- 
placed by  two  others  moro  simple, 

iy"— 1=0,  y+i=o. 

4.  Finally,  when  tho  equation  is  y-"  =  —  1,  or  y""-\-l=0,  we  know  that  tlio 
even  powors  of  real  quantities  will  always  give  positive  results ;  wo  hence 
conclude  that  all  tlio  roots  aro  imaginary.  Taking  y-  =  z,  tho  equation  reduces 
to  the  degree  n,  and  becomes  simply  ;"=  —  1. 

303.  I  now  proceed  to  determine  the  solutions  of  the  equations  ym  — 1  =  0, 
ym-\-l=z0,  in  some  particular  cases. 

Letm=2;  tho  equations  to  be  resolved  are 

;//  — 1=0,  whence  7/=il  ; 

i/* -)-l=0,  whence  y=i  \/ — 1. 
Letm=3;  to  resolve  the  equation  y — 1=0,  observe  that  it  has  for  a  root 
V  =  l;  we  divide  it.  by  y  —  1,  and  it  becomes 

y+y+1~Q>  wh '"  y= 5 • 

Hence,  the  three  roots  are 


_l+V— '*         -I--/— •: 

?/  =  !-?/  = 77 .2/=—         ,        -• 

If  we  take   the  equation   if -\-  I  =  I),  W6   shall  oh  BrVe  that   its   rooti  are  tho 

samn,  except  as  regards  Bign,  with  those  of  y  — i=u;  consequently,  they 

will  bo 

1_V^3  1-f  ,/_:!    ■ 

y=-hy= 5 ,3/  = 


BINOMIAL  EQUATIONS.  377 

Lot  m=4  ;  tho  equation  yi  — 1=0  may  bo  decomposed  into  two  others, 
y* — 1=0,  7/24-l=0;  and  from  these  equations  we  dorivo  tho  four  roots 

?/±l,?/±  V^l- 
The  equation  yf-\-l  will  be  resolved  differently;  by  adding  2if  to  both 
membors  of  tho  equation,  wo  can  write  it  thus: 

(^+1)8=2^; 
wo  can  then  decomposo  it  into  two  others, 

2/2+ 1 —y  V^,  tf+ 1  =  —  v  \/2 ! 

and,  finally,  from  those  wo  dorivo  tho  four  valuos  of  y, 

y  =  ±y/2±yV=<l,y  =  -lV2±lV^2. 
"Wo  could  havo  treated  tho  equation  2/1-j-l==0  ^  n  "reciprocal  equation. 
We  might  havo  observed,  also,  th;il;  it;  gives  y"=zh  V — *»  nn(t  tnati  taking 
successively  +-/ — 1,  —  a/ —  1,  wo  havo 

1=  ±V  +  l/^li  V=  i V  -  V^-L- 


V- 


We  havo  then  only  to  rcduco  theso  values  to  tho  form  a-4-/3-\/  —  1  by  tho 
process  in  Art.  104. 

By  raising  tho  equation  ;ym=pl=0  successively  to  tho  10°  degreo,  we 
shall  find  that  its  resolution  depends  on  that  of  tho  preceding  cases,  or  on  the 
resolution  of  reciprocal  equations,  which  reduce  it  to  a  degreo  less  than  tho  5°. 

Lot  us  examino,  first,  tho  odd  degrees.  If  wo  havo  the  equation  ?/"•  — 1=0, 
having  observed  that.it  has  tho  root  y  =  l,  wo  divide  it  by  y  —  1  ;  it  then  bo- 
comes 

a  reciprocal  equation,  which  we  shall  reduce  to  tho  2°  degreo.  To  do  this, 
wo  first  write  it  undor  tho  form 


(^)  +  (!/  +  J)  +  l=0. 


Then  take  t/+-=z,  which  gives  y--\-—=z~— 2;   and,   consequently,   the 

y  y 

equation  in  y  will  bo  changed  to  tho  following  : 

—  1±a/5 
22-}-z  — 1=0,  whence  z  = . 

Theso  valuos  being  known,  thoso  of  y  will  bo  by  tho  relation  y-\-—=z,  for 
this  relation  gives  


z4-  Vz2— 4 

y= § ; 

mid  Ave  have  only  to  substitute  instead  of  z  successively  each  of  its  two  values, 
in  order  to  find  the  four  imaginary  values  of  y.     Wo  havo  then  the  fivq  values 

of  y, 

y=l, 


-1+V5     V10  +  2V5    — 

y=--4 ± r    -V-i, 

-1--/5  .  V10-2V5    — - 
y= -. ± 7l V-l. 


378  ALGEBRA. 

The  equation  if  —  1=0  will  lead  to  the  ematicn  ;'-(-::  — "2:  — 1=0,  and 
the  equation  if  — 1  =  0  to  the  equation  ;44-;3— 3;:  — 2z-f-l=0. 

The  equations  i/5-|-l=0,  i/7-|-l  =  0,  t/°+  1=0  have,  except  as  regards  the 
signs,  the  same  roots  as  if  their  second  terms  had  been  — 1. 

Let  us  examine  the  even  degrees.     The  equations  f — 1  =  0,  if — 1  =  0, 
y10  — 1=0  do  not  offer  any  difficulty,  because  each  of  them  can  be  decom 
posed  into  two  others  whose  roots  are  known. 

Taking  -4-1  instead  of — 1,  the  analogous  equations  are 

2/6+1  =  0,  whence  y=\  V  — li 
2/  8-l~  1  =0,  whence  2/=\/  V  —  1, 

,  y°+ 1  =  0,  whence  y=yj  fy —1. 

But  we  know  the  values  of  fy  —  1,  \f  —  1,  %J  —  \  ;  we  have,  then,  only  to 
extract  the  square  roots  by  the  processes  in  Art.  104.  But  it  will  be  simpler 
to  treat  these  equations  as  reciprocal  ;  for  tho  transformed  equations  in  z,  on 
which  they  depend,  have  roots  which  aro  real,  and  are  veiy  easy  to  resolve. 

We  add  some  propositions  upon  binomial  equations,  preparatory  to  giving  a 
general  method  for  solving  those  of  all  degrees. 

PROPOSITION'    I. 

304.  If  a  be  one  of  the  imaginary  roots  of  tho  equation  .rn — 1=0,  then  any 
power  of  a  will  be  also  a  root. 

For,  since  a  is  one  root  of  the  equation  .rn  — 1=0,  therefore  an  =  l,  and,  con- 
sequently, 

a"-"  =  l,  a3n=l,  a4n  =  l,  &c,  also  a-"  =  l,  cr^ssl,  a~Sn  =  l,  &C, 
the  values  « 

a,  a2,  a3  . . . .,  a-1,  a~°,  a~3,  . . . ., 
thus  satisfying  the  conditions  of  the  equation,  are  roots  of  it. 

Corollary  1. — It  hence  appears  that  tho  roots  of  tho  equation  in  — 1=0  may 
be  represented  under  an  infinite  variety  of  forms,  each  term  in  the  following 
series  being  a  root,  viz. : 

a-3,  a-=,  a~\  1,  a,  a5,  a3, a"-',  an,  c"+l, a8n,  a=n+', 

in  which  series,  however,  there  can  not  be  more  than  n  quantities  essentially 
different,  otherwise  tho  equation  would  have  more  than  «  roots. 

PROPOSITION   II. 

305.  If  a  be  one  of  tho  imaginary  roots  of  tho  equation  xn-f-l=0,  then  any 
odd  power  of  o  will  be  also  a  root. 

For,  since  a  is  ono  root  of  the  equation  .<'  =  — 1.  therefore  an=  —  1  ;  and. 

since  every  odd  power,  whether  positive  or  negative,   of  — 1  is  also  — 1. 

therefore, 

n3"=— 1,  a'">=  — 1,  «•"=_  1,  &C, 

also 

a~3n=  — 1,  a-'r"'=— 1,  a-7n=—  1,  \'c.  ; 

so  that  tho  quantities 

a,  a3,  a-' ,  «-',  a~*,  <i    '• , 

arc  runts  of  the  equation.     These  roots,  therefore,  assume  an  infinite  variety 
of  forms,  although  there  can  nol  be  more  than  n  essentially  different. 


BINOMIAL  EQUATIONS.  379 

PROPOSITION    III. 

300.  To  determino  the  roots  of  the  equation  xn —  1  =  0,  wh&n  n  is  tho  square 
•f  a  prime  number  p. 

Put  xp=z,  then  xp — 2=0,  and  zp  — 1  =  0,  and  let  the  roots  of  this  last  equa- 
tion be  1,  (3,  /32,  /?3,  ....  /3p_1  ;  then,  by  substitution, 

'af— 1  =0, 

xP-Z=\X?-P=°> 

]   2-P  —  /?2  =  0, 

&c.     &c. 
Hence  the  pp  values  of  x,  in  these  p  equations,  will  evidently  be  all  difTerent, 
and  will  be  the  roots  of  the  equation  xpp — 1  =  0. 

To  determine  these  roots,  it  will  bo  sufficient  to  advert  to  Art.  300,  which 
proves  that  the  roots  of  xp — (3=0  are  equal  to  the  roots  of  x? — 1=0  multi- 
plied by  y[i  ;  and,  in  a  similar  manner,  the  roots  of  xp — |32=0  are  equal  to  the 
roots  of  xp — 1  =  0,  multiplied  by  tyj32,  &c. ;  therefore,  we  immediately  con- 
clude that  tho  roots  of 

xp-1  =0arel,  /?,  (3*,  (33 (3^  ) 

xp _/?  =0     ycpypipyii /3P-1  vp  [  -  un10  "  ™°™  ot 

tp  -13* = 0  Vt3\  [3  ys3\  (3'  Vi32  •  •  •  •  Z3""1  VP  >       * 

&c.  &c.  &c. 

For  example,  let  it  bo  required  to  find  the  roots  of  x9 — 1  =  0. 
The  roots  of  x3 — 1=0  are 


-1+  V-3    -l-V-3 
7  -2  '  2 

hence  the  roots  of  x'J — 1=0  are 


.     -l+V-3    -l-V-3 


l+y_3    -l+V-3, ,-l+V-3 
v « > o V  o  i 


_l_V-33    -1+^-3    3/_l-V 


v — v — .  v- 


-1+ V-3      -1--/-3    -l-y-3      -l-V-3 
o  V  o  '  o  V  o 

iv  <6  <6  iv 

The  foregoing  propositions  have  been  devoted  chiefly  to  an  examination  of 
the  properties  and  relations  of  these  roots,  and  not  to  the  actual  exhibition  of 
their  values,  although,  as  in  the  proposition  above,  one  or  two  examples  of  the 
solution  have  been  given  to  illustrate  the  reasoning.  To  obtain  the  imaginary 
roots,  however,  in  their  simplest  form,  that  is,  in  the  form  a+6  V — 1>  and 
for  all  values  of  the  exponent,  requires  the  aid  of  a  theorem,  borrowed  from 
the  science  of  Trigonometry. 

307.  The  theorem  to  which  we  refer  is  the  well-known  formula  of  De 
Moivre,  given  in  most  books  on  Analytical  Trigonometry. 

(cos  a+  sin  a  .  V — l)n=  cos  na+  sin  na  .  V  — 1 ; 
which,  if  the  arc  2lcn  (tt  being  a  semi-circumference,  and  k  any  integer)  be 
substituted  for  na,  becomes 


380  ALGEBRA. 

2&t  ,  2kn       . ,  .  

(cos ±  sin .  V  — 1)"=  cos  2/cTTztz  sin  2/c~  .  y  — 1  ; 

v         n  n       y         > 

that  is,  since 

cos  2ktt=1,  and  sin  2/c~=0, 

nJcn  2/,'t        

(cos  - — ±  sin .  V  — 1)"  =  1  ; 


so  that  the  expression 


2kn  .  licit 


cos i  sin .  y/  — 1, 

n  n 


couiprehends  in  it  all  the  n  roots  of  unity,  or  all  the  particular  values  c£  x 
which  satisfy  the  equation  xn — 1  =  0. 

Although,  in  this  general  expression,  the  value  of  k  is  quite  arbitrary,  yet 
assume  it  what  we  will,  the  expression  can  never  furnish  more  than  n  differ- 
ent values.     These  different  values  will  arise  from  the  several  substitutions  of 

up  to  the  number  — - — ,  inclusively,  if  n  is  odd,  and  up  to  -,  if  n  is  even  ;  and 

for  substitutions  beyond  these  limits  the  preceding  results  will  recur.     To 
prove  this,  let  us  actually  substitute  as  proposed  ;  we  shall  thus  have  the  fol 
lowing  series  of  results,  viz. : 


for  k  =  0  ....  .r=  cos  0    i  sin  0    .  ■/  — 1=1 

2k        .    2w       . 

k  =  l  ....  x=  cos  — ±  sin  —  .  V — 1 

n  n 

4t  4~        

A:=2 x=  cos  — ±  sin  —  .  V — 1 

n  n 

Qk  .  6k 


£=3  ....  .t=  cos  — rt  sin  —  .  -J — I 

n  n 

„_1                        (n-l)T  ,      .    (n-l)ff        — 
fc  = — -—  .  .  xss  cos  ±  sm .  v — 1- 

2  n  n 

Each  of  these  expressions,  except  the  first,  involves  two  distinct  values  :   g 

that,  omitting  the  value  given  by  k =0,  there  are  n — 1  values,  and,  consequent 

ly,  altogether,  there  are  n  values ;  and  that  they  are  all  different  is  plain,  be 

cause  the  area 

2k   4k   6k  (n  —  1)t 

u,       ,      ,       ,  .  •  •  •  ~~  , 

7i     n     n  n 

being  all  different,  and  loss  than  it,  have  all  different  i  Tin  arcs  which 

would  arise  from  continuing  the  substitutions  aro 

(«4-1)tt    (;i+3)x    (n+3)ir 
n      '        n      '        n 
or,  which  aro  the  same, 


-,  &c. ; 


(W-1)t  (w-3)y  (w-5)x 

r— ,  2tt ,  2ir ,  cVc, 

7i  71  n 

and  the  sines  and  cosines  of  theso  are  respectively  tin-  same  us  the  sines  ind 
of 1 1  io  arcs 


BINOMIAL  EQUATIONS.  381 

(n — l)n   (n  —  3)?r   (n  —  b)n 

n  n  n 

which  have  already  occurred.* 

w 
If  n  is  an  even  number,  the  final  substitution  for  k  must  be  -  instead  of 

* i 

■  0    ,  as  above ;  and,  therefore,  the  final  pair  of  conjugate  values  for  x  will  be 


x=  cos  ffi  sin  7r  .  -/  —  1  =  —  1, 
which  values  of  x  differ  from  all  the  other  values,  because  in  them  no  arc  oc- 
curs so  great  as  n. 

The  arcs  which  would   arise  from  continuing  the   substitutions   beyond 


j      n 
kz=i-  are 


(n-\-2)n   (n+4)?r   (n+G)ir 

— ,  &c. ; 


n  n  n 

or,  which  are  the  same, 

(n — 2W            (n — 4)tt            (n — 6)rr 
2tz—- '-,  2n— -,  2?r— -,  &c, 

74  74  71 

and  the  sines  and  cosines  of  these  are  respectively  the  same  as  the  sines  an<. 
cosines  of  the  arcs 

(n — 2)k   (n — 4)xr  (n — 6)k 

,  ,  ,    (.VC.j 

n  n     ,         n 

which  have  already  occurred.* 

It  is  easy  to  see  that  in  every  pair  of  conjugate  roots,  each  is  the  reciprocal 
of  the  other.     In  fact,  whatever  be  k, 

2kn  2Z.-7T        ; ,  2&7T  27ctt         . 

(cos 4-  sm .v  — 1)  (cos —  sin .  v  — 1)  = 

v         n     '  7i  '  v         7i  n  ' 

2kn              2kir 
cos2 4-  sin2 =1, 

J  71       '  71 

which  shows  that  the  two  factors  in  the  first  member  are  of  the  form  a,  -. 

a 

We  have  proved  (Art.  304)  that  every  power  of  an  imaginary  root  of  the 

binomial  equation  is  also  a  root ;  but,  unless  n  be  a  prime  number,  wo  could 

not  infer  that  all  tho  roots  would  ever  be  produced  by  involving  any  one  of 

them.     Such  would  not,  indeed,  be  the  case.     There  is  always,  however, 

one  among  the  imaginary  roots  of  which  the  involution  will  furnish  all  the 

others;  it  is  the  first  imaginary  root,  or  that  due  to  the  substitution  k=l,  in 

the  foregoing  series  of  values ;  for,  by  De  Moivre's  formula,  the  powers  of  this 

produce  all  the  others,  thus  : 

2ir  2t        47T  47r 

(cos  — -j-  sin  —  .  -J  —  1;-=  cos  — 4-  sin  —     -'  — 1 
7i   '  n  '  n   '  n 

6tt  6t 


(cos  — +  sin  —  .  V  —  iy=  cos  — +  sin  —  .  V  —  1 

v        7i   '  n  '  n   '  n 


2tt        .    2tt        , !!zi  n— 1  n— 1 

(cos  — 4-  sm  —  .  V  — 1)  -  =  cos 71-4-  sin .  v  — 1. 

*  Tlie  signs  of  the  sines  will,  however,  be  different;  but  the  only  effect  of  this  difference 
J  evidently  to  furnish  each  pair  of  conjugate  roots  in  an  inverse  order. 


o82  ALGEBRA. 

These  powers  of  the  first  imaginary  root,  which  we  may  cal  a,  th.is  fur- 
nish one  half  of  the  entire  number  of  imaginary  roots,  and  the  reciprocals  of 
these  being  the  other  half,  all  of  them  are  determined  from  the  first ;  the 
imaginary  roots  are,  therefore, 

n— 1 


o, 

a2,  a3, 

1 

1     1 

1 

a 

a»'  a3' 

When  n  is  even,  the  last  power  will  be 


Mn  Mrc 


(cos  — 4-  6in  —  .  \/  — lYJ=cos  tt+  sin  n  .  J  — 1 ; 
and  the  imaginary  roots  are,  therefore, 

n 

a,  as,  a3,   ....  a* 
111  1 

a'  ^  *5 1 

o? 

308.  By  the  general  formula  (Art.  307),  we  are  enabled  to  determine  all  tLo 

roots  of  the  equation 

xn  +  1=0  ; 

for,  since 

cos-(2&+l)7r=—  1,  and  sin  (2&4-l)7r=0, 

that  formula  gives 

2/fc+l     ,  2fc+l         . 

(cos tt±  sin n  .  \/  — l)n  = 

n  n  ' 


cos  (2A:+l)7r±  sin  (2fc-fl)jr  -V—  las— 1; 
hence  the  n  values  of  x  are  all  comprised  in  the  general  expression 

2&+1    '       .    2&-fl 


x=  cos  ttJL  sin  7r  .  -/  — 1 ; 

ft  71 

which,  by  putting  for  k  the  values  0,  1,  2,  3,  &c,  in  succession,  furnishes  the 
following  series  of  separate  values,  viz. : 


n  .  7T 


for&=0  ....  .t=  cos-i  sin  -  .  J — 1 

n  n 

3ir"         .     3rr       . 

«=1  ....  t=  cos  — ±  sin  —  .  /  — 1 

n  n 

5t  .  5t 


&=2  ....  x=  cos  — i  sin  —  .  -J  —  1 

n  n     v 


,      «— 1  .      •  ,- — 

Jc= — - —  .  .  x=  cos  n±  sin  t  .  -/  — 1  =  — 1  ; 

or,  when  n  is  even, 

71  —  2  /        ~\  ir 

jfcsss — ; —   .   .   .  .T=  cos  1 7r — -J    -  sin  (rr — -  .  -J — 1). 

Now  that  the  foregoing  system  of  n  roots  aro  all  different  is  obvious,  siiK* 

7T   3k  5tt  tjt  tt 

- ,  — ,  —  ....  — ,  or  n— -, 
n    n     n  n  n 

are  all  different  arcs,  of  which  the  greatest  does  not  exceed  a  serai-circum 


BINOMIAL  EQUATIONS.  383 

ference.     If  the  preceding  series  be  extended,  it  will  be  easy  to  prove,  after 

what  has  been  done  in  Art.  307,  that  the  values  formerly  obtained  will  recur. 

As  in  the  former  case  of  the  general  problem,  so  here,  each  root  may  be 

derived  from  the  first  pair  of  the  series ;  thus,  denoting  the  first  root,   cos 

7T  W 1 

-rt  sin  -  .  -/  —  1,  by  a  or  -,  according  as  the  upper  or  lower  sign  is  taken, 

TV  Tt  Q> 

wo  evidently  have,  for  the  preceding  series,  the  following  equivalent  expres 
sions,  viz.  : 

a,  a3,  a5,  ....  a"  -\ 

111  1  >  when  n  is  odd, 

a'  a?'  tf a"  ) 

and 

a,  a3,  a5, a"-1  \ 

111  1    (  when  n  is  even, 

a'  a3'  ~afi a^  ) 

For  further  researches  on  tho  theory  of  binomial  equations,  the  student  may 
consult  Lagrange's  Traite  de  la  Resolution  des  Equations  Numeriques,  Note 
14  ;  Legendre's  Theorio  des  Nombres,  Part  V.  ;   the  Disquisitiones  Arith- 
meticae  of  Gauss ;  Barlow's  Theory  of  Numbers  ;  and  Ivory's  article  on  Equa 
tions,  in  the  Encyclopaedia  Britannica. 

309.  We  have  already  frequently  had  occasion  to  notice  multiple  values  of 
radicals,  without  fixing  the  precise  number  which  might  exist,  except  for  rad 
icals  of  the  second  degree.     It  is  time  to  introduce  the  following  proposition  : 

Every  radical  has  as  many  values  as  there  are  units  in  its  index,  and  has 
no  more  ;  in  other  words,  every  quantity  has  as  many  roots  of  a  given  degree 
as  there  are  units  in  the  index  of  that  degree. 

If  the  given  radical  be  represented  by  the  general  form  \/A,  this  radical 
designates  evidently  all  the  quantities,  real  or  imaginary,  which,  raised  to  the 
power  m,  reproduce  A  ;  consequently  they  are  merely  the  values  of  x  in  the 
equation  arm=A.  But  we  know,  from  the  general  theory  of  equations,  that 
every  equation  of  tho  mtb  degree  has  m  values  of  the  unknoAvn  quantity,  which 
will  each  satisfy  it ;  hence  the  proposition  is  proved. 

This  will  serve  to  explain  some  paradoxes.  Let  there  be  the  expression 
May/ — 1.  By  reducing  the  second  radical  to  the  index  4,  it  becomes 
y(  —  Yf,  and  the  given  expression  reduces  to  \Ja,9.  result  which  might  be 
supposed  absurd,  because,  a  being  positive,  the  result  represents  a  real  quan- 
tity, while  the  proposed  expi'ession  appears  to  be  imaginary. 

There  is  here  a  confusion  of  ideas.  If  in  the  expression  tya  V  —  1  tno 
radical  is  an  arithmetical  determination,  it  is  true  that  this  expression  is 
imaginary  ;  but  if  V a  be  taken  in  all  its  generality,  and  we  represent  it  by  a' 
multiplied  by  the  four  roots  of  unity,  or 


a',  —a',  a'  yf  —  I,  —a'  y  — 1, 


we  perceive  that  some  of  these  values  of  V«,  multiplied  by  *J — 1,  cause  this 
imaginary  factor  to  disappear,  and  the  proposed  expression  becomes  real. 

I  shall  terminate  this  article  by  the  explanation  of  a  paradox  which  presents 
itself  in  the  employment  of  fractional  exponents.     Let  there  be  the  expres- 

2  3  L 

sion  af.     If  the  fraction  £  be  simplified,  the  expression  a*  becomes  a-.    Then, 

in  repassing  to  tho  radicals,  we  have  t/a-=  ^a.     This  equality,  however   is 


384 


ALGEBRA. 


not  wholly  true,  because  the  first  member  has  four  values,  and  the  second 
but  two. 

The  difficulty  may  be  presented  in  a  general  manner  by  placing 

mp  m 

and  in  concluding  from  thenco  that 

To  discover  the  cause  of  this  error,  we  must  remember  that  the  fractional 
exponent  is  but  a  convention,  by  means  of  which  we  express  in  another  way 
that  the  root  of  a  certain  power  is  to  be  extracted,  and,  therefore,  this  expo- 
nent must  not  be  regarded  in  the  light  of  an  ordinary  fraction. 

TILE  DETERMINATION  OF  THE  IMAGINARY  ROOTS  OF  EQUATIONS. 

310.  In  what  relates  to  the  limits  of  roots  at  Art.  283  and  following,  real  roots 
only  were  in  view.  We  shall  show  here  how  the  limits  may  be  obtained 
for  the  moduli  of  all  roots,  whether  real  or  imaginary.  Let  us  consider  the 
equation 

a:ra+Pxm-1  +  Qxro-3...=0 (1) 

in  which  P,  Q . . .  may  be  real  or  imaginary.  In  order  that  a  value  of  x  may 
be  a  root,  it  is  necessary  that,  after  having  substituted  it  in  the  result,  the 
modulus  should  be  zero. 

Call  v  the  modulus  of  x,  and  p,  q, . .  .  those  of  the  coefficients  P,  Q Ac- 
cording to  Art.  239,  those  of  the  terms  of  the  equation  will  be  vm,  pvm~\ 

qvm~\  . .  .,  and  that  of  the  part  Pa^+Q^-H can  not  surpass  the  sum 

„rm-i_i_oVm-2  t .  t     Then,  if  we  choose  for  v  a  value  1  such  that  we  have 

v™—pV™-i—qvm-^ =0,  or  >0  .  .  .  .  (2) 

we  are  sure,  by  virtue  of  the  article  just  cited,  that  the  modulus  of  the  first 
member  of  the  equation  (1)  will  not  be  less  than  the  above  difference  ;  and  that 
from  this  point  the  modulus  will  not  be  zero,  or,  what  is  the  same  thing,  the 
value  substituted  in  place  of  x  will  not  be  a  root  of  the  equation.  Every  value 
of  v  above  "K  will  render  this  difference  greater ;  then  1  is  a  superior  bmit  of 

the  moduli. 

The  quantity  "k  will  be  always  easy  to  determine,  because  it  will  be  sufficient 
to  substitute  in  the  difference  (2)  in  place  of  v,  increasing  positive  values  until 
this  difference  becomes  positive.  If  tho  coefficients  P,  Q . . .  are  real,  the 
moduli  p,q,---  will  be  these  coefficients  themselves,  but  taken  positively ;  and 
if  we  designate  tho  greatest  of  these  values  by  N,  we  can  take  at  once  for  die 
superior  limit  A=N-f-L 

To  have  an  inferior  limit,  wo  make  ar=-,  determine  in  the  transformed  in  y 

the  superior  limit  of  the  moduli  of  the  roots,  and  finally  divide  unity  by  this 
limit. 

311.  It  has  already  been  proved  that  imaginary  roots  always  enter  into 
equations  in  conjugate  pairs  of  the  form  a±i3i/—l.  And  this  previous 
knowledge  of  tho  form  which  ovory  root  musi  take  suggests  a  method  for  the 
actual  determination  of  the  proper  numerical  values  for  a  ami  >'  in  any  proposed 
case.     The  method  is  as  follows  : 

Let  .r"  +  A0-,.r"  lH Aj  +  N=0 


IMAGINARY  ROOTS  OP  EQUATIONS.  38* 

be  an  equation  containing  imaginary  roots;  then,  by  substituting  n-f-JV — * 
for  x,  we  have 


(a+3V-l)"  +  An-1(«+;W-ir1+..A(a+/3V-l)  +  N  =  0; 
or,  by  developing  the  terms  by  the  binomial  theorem,  and  collecting  the  real 
and  imaginary  quantities  separately,  we  have  the  form 

M  +  NV-1=0, 
an  equation  which  can  not  exist  except  under  the  conditions 
M  =  0,  N  =  0 (1) 

From  these^wo  equations,  therefore,  in  which  M,  N  contain  only  the  quan- 
tities a,  /•,  combined  with  the  given  coefficients,  all  the  systems  of  values  of  a 
and  j3  may  bo  determined  ;  and  these,  substituted  in  the  expression  a-\-fi  -\/  — I, 
will  make  known  all  the  imaginary  roots  of  the  proposed  equation ;  those  that 
are  real  corresponding  to  fS=0. 

It  is  obvious  from  the  theory  of  elimination  as  developed  at  page  1-37,  and 
from  the  method  of  numerical  solution  explained  in  Art.  255,  that  the  labor  of 
deducing  from  this  pair  of  equations  the  final  equation  involving  only  one  of  the 
unknowns  a,  /?,  and  of  afterward  solving  the  equation  for  that  unknown,  will 
in  general  be  veiy  laborious  for  equations  above  the  third  degree.  Lagrange, 
by  combining  with  the  principle  of  this  solution  the  method  of  the  squares  of 
the  differences  explained  at  Art.  278,  avoids  both  the  elimination  and  subse- 
quent solution  here  spoken  of.  It  is  easy  to  see  how  this  may  be  brought 
about  if  we  have  any  independent  means  of  determining  one  of  the  unknowns 
0:  for  the  adoption  of  these  means  would  enable  us  to  dispense  with  the  elimi- 
nation ;  and  as  the  substitution  of  the  valuo  of  (3  in  both  of  the  equations  (1) 
would  convert  those  equations  into  two  simultaneous  equations  involving  but 
one  unknown  quantity,  their  first  members  would  necessarily  have  a  common 
factor  of  the  first  degree  in  a,  which,  equated  to  zero,  would  furnish  for  a  the 
proper  value  to  accompany  (3 ;  and  thus,  instead  of  solving  the  final  equation 
referred  to,  we  should  only  have  to  find  the  common  measure  between  the 
two  polynomials  M,  N  containing  the  unknown  quantity  a. 

Now  corresponding  to  every  pair  of  imaginary  roots  a-\-(3  -<J  — 1,  a — {3  \/  —  1, 
there  necessarily  exists,  in  the  equation  of  the  squares  of  the  differences,  a 
real  negative  root  — 4/3*;  so  that  if  all  the  negative  roots  of  the  latter  equation 
be  found,  the  quantity  — 4/33  must  appear  among  them  ;  from  which  the  value 
of /3  would  bo  immediately  obtained,  and  thence,  by  aid  of  the  common  meas- 
ure as  just  explained,  tho  corresponding  value  of  a. 

But  the  equation  of  the  squares  of  tho  differences  may  have  a  greater  num- 
ber of  negative  roots  than  there  are  pairs  of  imaginary  roots  in  the  proposed  : 
which,  however,  can  not  happen  except  two  non-conjugate  imaginary  roots  have 
equal  real  parts,  or  except  a  real  root  be  equal  to  the  real  part  of  an  imaginary  * 
root.  Lagrange  discusses  these  peculiarities,  and  establishes  the  exactness 
and  generality  of  tho  principle  in  question,  as  follows  : 

When  the  real  parts,  a,  y,  &c,  of  the  imaginaries 


n-r-jS -v/ — 1'  a—  PV—  1 

&c.  6cc. 

are  unequal,  as  well  when  compared  with  one  another  as  when  compared  with 
the  real  roots  a,  b,  c,  &c,  it  is  evident  that  the  equation  of  the  squares  of  tho 

Bb 


386  ALGEBRA. 

differences  can  not  have  any  other  negative  roots  than  th:6e  furnished  by  the 
several  pairs  of  conjugate  imaginary  roots,  and  which  are  . 

_40»,  —  4cP,  &c. 
All  the  other  roots,  not  arising  from  the  differences  furnished  by  the  real 
roots,  a,  b,  c,  &c,  will  evidently  be  imaginary ;  those  between  the  real  and 
imaginary  roots  supplying  the  forms 

(a-b  +  pV-l)\  {a—h—pV—iy 
&c.  &c. 

and  those  between  the  non-conjugate  roots  tho  forms 


\(a-7)  +  ^-6)V-lL\"-,  \(a-})-(3-6)V-l  ' 
U*-7)  +  iP+8)V-l\*,  \(a-y)-(3+6)V-l\* 
so  that  in  this  case  every  negative  root  in  the  auxiliary  equation  will  indicate  a 
pair  of  imaginary  roots  in  the  proposed,  and  will,  moreover,  supply  tho  value 
of  the  imaginary  part.  But  if  it  happen  that  among  tho  quantities  a,  y,  occ, 
thero  be  found  any  equal  among  themselves,  or  equal  to  any  of  the  quantities 
a,  b,  r,  &c,  then  the  auxiliary  equation  will  necessarily  have  negative  roots, 
corresponding  to  which  there  can  be  no  imaginary  pair  in  the  proposed  equa- 
tion. 

For  let  a=a,  then  the  two  imaginary  roots  (a— a-\-(iy/ — l)3,  (a — a — /3 
-/ — 1):  will  become  — /J2  and  — /?»,  and,  consequently,  real  and  negative  ;  so 
that  if  tho  proposed  equation  contain  only  two  imaginary  roots,  a-\-(3  y/  —  1  and 
a — j3  \/  — 1,  then,  in  the  case  of  a=a,  the  equation  of  the  squares  of  the  differ- 
ences will  contain,  besides  the  real  negative  root  — 4/3°,  the  two  — j9»,  — CP, 
both  negative  and  equal. 

We  thus  see  that  when  the  equation  of  the  squares  of  the  differences  has 
three  negative  roots,  of  which  two  are  equal  to  one  another,  the  proposed  may 
have  either  three  pairs  of  imaginary  roots,  or  but  a  single  pair. 

If  the  proposed  contains  four  imaginary  roots,  a+/3-\/  —  1,  a  —  3  -y/ — 1, 
y+^V —  1,  y —  (5\/ — 1,  then  tho  equation  of  the  squares  of  the  differences 
must  contain  tho  two  negative  roots  — 4  -'-  and  — 4cP;  if  a=<7,  it  must  also 
contain  the  two  equal  negative  roots  — /J2,  — /J2 ;  and  if,  moreover,  y=b,  it 
must  contain,  in  addition  to  these,  the  negative  pair  — 62,  — <5-;  and  lastly,  if 
assy,  the  four  imaginary  roots 

j(a-y)  +  (/?-H)V-lp,  f(a->)-(^+<5)V-lj* 
will  be  converted  into  the  two  negath  o  pairs 

-(/i—5)2,  -(/?-d)»;    -(0+«f)»,  -(,-3+<5)-\ 
Hence  we  may  deduce  tho  following  conclusions,  \i/.. : 
(1)  When  all  the  real  negative  roots  of  tho  equation  of  tho  squares  of  the 
differences  are  unequal,  then  tho  proposed  will  necessarily  have  so  many  pairs 
of  imaginary  roots. 

If  in  this  case  wo  call  any  one  of  these  negative  roots  — w,  wo  ahaD  have 

■\f w 
fl=— —  ;  and  if  this  value  be  substituted  fur  >'  in  the  two  equations  (1).  and  the 

operation  for  tin-  common  measure  of  their  6rsl  members  be  carried  on  till  we 
krnve  at  a  remainder  of  tho  first  degree  in  a,  tho  proper  value  of  a  will  be  ob 


IMAGINARY  ROOTS  OF  EQUATIONS.  3£7 

♦ained  by  equating  this  remainder  to  zero.  Thus,  each  negative  root,  — u\ 
will  furnish  two  conjugate  imaginary  roots,  a-\-(3  V  — 1,  and  o — (3  y/  — 1. 

(2)  If'among  the  negative  roots  of  the  equation  of  the  squares  of  the  differences 
equal  roots  arc  found,  then  each  unequal  root,  if  any  such  occur,  will,  as  in 
the  preceding  case,  always  furnish  a  pair  of  imaginary  roots.  Each  pair  of 
equal  roots  may,  however,  give  either  two  pairs  of  imaginary  roots  or  no  im- 
aginary roots,  so  that  two  equal  roots  will  give  either  four  imaginary  rpots  or 
none  ;  three  equal  roots  will  give  either  six  imaginary  roots  or  two ;  four  equal 
roots  will  give  either  eight  imaginary  roots,  or  four,  or  none  ;  and  so  on. 

Suppose  two  of  the  negative  roots,  — w,  — w,  are  equal ;  then  putting,  as 

■\/  w 
above,  /?=——,  we  shall  substitute  this  value  of /3  in  the  two  polynomials  (1), 

and  shall  carry  on  the  process  for  the  common  measure  between  these  poly 
nomials  till  we  arrive  at  a  remainder  of  the  second  degree  in  a  ;  since  the  poly 
nomials  must  have  a  common  divisor  of  the  second  degree  in  a,  seeing  that  the 
equations  (1)  must  have  two  roots  in  common,  on  account  of  the  double  value 
of/?. 

Equating,  then,  this  quadratic  remainder  to  zero,  we  shall  be  furnished  witn 
two  values  for  a :  these  may  be  either  both  real  or  both  imaginary.  In  the 
former  case  call  the  two  values  a'  and  a" ;  we  shall  then  have  the  four  imagin- 
ary roots 


a'+pV—l,  a'~ PV  —  li  o"+j3V—  1,  a"—  /?•/— 1. 

In  the  second  case,  the  values  of  a  being  imaginaiy,  contrary  to  the  condi- 
tions by  which  the  fundamental  equations  (1)  are  governed,  we  infer  that  to 
the  equal  negative  roots  — w,  — w,  there  can  not  correspond  any  imaginary 
roots  in  the  proposed  equation. 

If  the  equation  of  tho  squares  of  the  differences  have  three  equal  negative 

roots,  — w,  — w,  — w,  then,  putting,  as  before,  /?=——,  we  should  operate  on 

the  polynomials  (1),  for  the  common  measure,  till  we  reach  a  remainder  of 
the  third  dogreo  in  a;  this  remainder,  equated  to  zero,  will  furnish  three  values 
of  a,  which  will  either  be  all  real,  or  one  real  and  two  imaginary.  In  the  first 
case  six  imaginary  roots  will  be  implied  :  in  the  second  only  two ;  the  imagin- 
ary .values  of  a  being  always  rejected,  as  not  coming  within  the  conditions  im- 
plied in  (1). 

It  follows  from  the  above,  and  from  what  has  been  established  in  Art.  259, 
that  there  are  at  least  as  many  variations  of  sign  in  the  equation  of  the  squares 
of  differences  as  there  are  combinations  of  two  real  roots  in  the  proposed 
equation.  Also,  it  must  have  at  least  as  many  permanences  of  sign  as  there 
are  pairs  of  conjugate  imaginary  roots  in  the  proposed  equation ;  or,  in  other 
words,  it  can  not  have  a  less  number  of  permanences  of  sign  than  half  the  num- 
ber of  imaginaiy  roots  in  the  proposed  equation. 

Hence  we  may  infer,  that  if  the  equation  of  the  squares  of  the  differences 
have  its  terms  alternately  positive  and  negative,  there  can  be  no  imaginary 
root  in  the  proposed  equation. 

The  foregoing  principles  are  theoretically  correct ;  but  the  practical  appli- 
cation of  them,  beyond  equations  of  the  third  and  fourth  degrees,  is  too  labo- 
rious for  them  to  become  available  in  actual  computation.     We  give  the  follow 
ing  illustration  of  them  from  Lagrange- 


388  ALGEBRA. 

312.  To  determine  the  imaginary  roots  of  the  equation 

r<  — 2.r— 5=0. 
Computing  the  equation  of  the  squares  of  the  differences  from  Hie  genera. 
formula  for  the  third  degree  at  Art.  279,  viz. 

2»-J-6pz9+9pBz+4p»+27g*=0, 
in  which  p  =  — 2  and  q  = — 5,  we  have 

z»— 12z»+36z+643=0. 
In  order  to  determine  the  negative  roots  of  this  equation,  change  the  alternate 
signs,  or  put  ;  =  — M>,  and  then  change  all  the  signs,  converting  the  equation 
into 

vfi+l2il?+3&W— 643=0, 
and  seek  the  positive  root,  which  is  found  by  trial  to  lie  between  5  and  6. 
Adopting  Lagrange's  development.  Art.  297,  this  root  proves  to  be 

*>=5+6  +  i     1 

6  +  ,  &c, 

from  which  we  get  the  converging  fractions  (see  Continued  Fractions) 

31    160   991 

5'  ~6~'  "3P  192'        ' 

■\/w 
Knowing  thus  an  approximate  value  of  ic,  wo  know  /3=  — — -. 


In  order  now  to  get  the  equations  (1),  p.  385,  substitute  a-\-{3^  —  1  for  xin 
the  proposed  equation,  and  form  two  equations,  one  with  the  real  terms  of 
the  result,  the  other  with  the  imaginary  terms  ;  we  shall  thus  have  the  equa- 
tions (1)  referred  to,  viz., 

o3— (3^+2)0—5=0 
3a2_/y>_2  =  0, 
in  which  (i  is  known. 

Seeking  now  the  greatest  common  measure  of  the  first  members  of  these 
equations,  stopping  the  operation  at  the  remainder  of  the  first  degree  in  a,  and 
equating  that  remainder  to  zero,  we  have 

15 
C=-8^H' 
and  thus  both  a  and  /?  are  determined  in  approximate  numbers. 

313.  There  is  another  method  of  proceeding  for  the  determination  of  im- 
aginary roots,  somewhat  different  from  the  preceding,  being  independent  of 
the  equation  of  the  squares  of  tho  differences.  It  is  suggosted  from  the  fol- 
lowing considerations  : 

Since  tho  quadratic,  involving  a  pair  of  imaginary  conjugate  roots,  is  always 
nf  the  form 

—  2ax+a2  4-/^  =  0, 
every  equation  into  which  such  roots  enter  must  always  be  accurately  divisible 
liy  a  quadratic  divisor  of  this  form;  that  is,  the  proper  values  of  a  and  |9  are 
s:icli  that  the  remainder  of  the  firsl  degree  in  r,  resulting  from  the  dh 
must  'oo  zero.  This  furnishes  a  condition  from  which  thoso  proper  values  of 
o  ami  /?  may  be  determined  ;  the  condition,  namely,  thai  the  remainder  spoken 
of,  A.r — B,  must  be  equal  to  zero,  independent  of  particular  values  of  .r;  and 
thii  implies  the  twofold  condition 


IMAGINARY  ROOTS  OF  EQU  VTIONS.  389 

A  =  0,  B  =  0, 
from  which  a  and  (i,  of  which  A  and  B  are  functions,  may  bo  determined. 
As  an  example,  let  the  equation  proposed  bo 

a;4_j_  4xs_j_  e2.2_|_  4^4.  5  _  o. 

Dividing  the  first  member  by 

tf—Zax+cP+p, 

we  have  for  quotient 

i-s  +.  (4  4. 2a)x-\-  6  +  8a  +  3a' — (P, 
and  for  the  remainder  of  the  first  degree  in  x 

(4+12a+12as4-4o3— 4a/3»— 4j3*)x— 

(o«+^)(6  +  8o+3oS—  j32)  +  5, 
which,  hoing  equal  to  zero  whatever  be  the  value  of  x,  furnishes  the  two  equa- 
tions 

4+12a-fl2aB+4o3— 4flj8»— 4j9s=0 

(o'+j8s)(6+8o+3o=— /P)  +  5=0. 

From  the  first  of  these  we  get 

/?*=(!  + ar- 
and  this,  substituted  in  the  second,  gives 

4a-<-f]  6a" -{-24a2 -f  16a=0, 
two  roots  of  which  are  0  and  —2 ;  the  other  two  are  imaginary,  and  must, 
consequently,  he  rejected  as  contrary  to  tho  hypothesis  as  to  the  form  of  th* 
indoterminate  quadratic  divisor. 

The  two  real  values  of  a,  substituted  in  the  expression  above  for  /32,  give 
fora=      0,  (T-  =  V  .-.  /3=+l 

a=-2,  /?*=(-l)'- .-.  /3=-l 

and,  consequently,  the  component  factors  of  the  original  quadratic  divisor,  viz.. 
the  factors 

x—a—Pi/—l,  x— a+/3-/^-li 
furnish  these  two  pairs  of  imaginary  roots,  viz., 

x=  V— i,  •?=—  V— i) 

and 


x=z— 2—  -/  — 1,  x=—  2+  V  —  1. 
This  method,  like  that  before  given,  is  impracticable  beyond  very  narrow 
limits,  because  of  the  high  degree  to  which  the  final  equation  in  a  usually 
rises.  And  it  is  further  to  be  observed  of  both,  and,  indeed,  of  all  methods 
for  determining  imaginary  roots  by  aid  of  the  real  roots  of  certain  numerical 
equations,  that  whenever,  as  is  usual,  these  real  roots  are  obtained  only  ap- 
proximately, our  results  may,  under  peculiar  circumstances,  be  erroneous. 
For  instance,  in  the,  two  methods  just  explained  we  have  two  equations, 
/(a)=0,  F(/3)=0,  where  the  coefficients  of  a  in  the  first  are  functions  of  ;. 
and  the  coefficients  of  (3  in  the  second  functions  of  a  ;  hence,  whichever  of 
these  symbols  be  computed  approximately,  in  order  to  furnish  determinate 
values  for  th#  coefficients  of  the  other,  these  coefficients  must  vary  slightly 
from  the  true  coefficients  ;  andv  consequently,  under  this  slight  variation  of  the 
coefficients,  real  roots  may  become  converted  into  imaginary,  and  imaginary 
into  real. 


390  ALGEBRA. 

The  terms  imaginary  and  impossible  have  been  thought  objectionable  when 
applied  to  the  roots  of  equations,  inasmuch  as  definite  algebraic  expressions 
are  always  possible  for  these  roots. 

\  specimen  of  a  strictly  impossible  equation  would  be  the  following: 

2z— 5+  -^—7=0, 


when  plus  before  the  sign  y/  implies  the  positive  root  ^x1  —  7.  No  ex- 
pression, either  real  or  imaginary,  can  satisfy  the  condition  or  represent  a  root 
of  this  irrational  equation. 

The  terms  imaginary  and  impossible,  when  used,  should  be  understood 
rather  as  applying  to  the  solutions  of  the  problem  from  which  the  equation  is 
derived  than  to  the  expressions  for  the  roots.  The  number  of  solutions  which 
the  problem  admits  will  ordinarily  be  expressed  by  the  degree  of  the  equa- 
tion, but  certain  suppositions  affecting  the  values  or  signs  of  the  coefficients 
may  cause  some  of  these  solutions  to  become  absurd  or  impossible,  and  these 
will  be  indicated  by  the  form  a-\-b  V — 1  for  the  roots,in  which  b  is  not  zero. 

THEORY  OF  VANISHING  FRACTIONS. 
314.  From  the  principles  established  in  (Art.  253),  we  readily  derive  the 
following  consequences,  viz.  : 
Since 

f(x)  =  {x—al)(x—a2)(x—a3){x—a,) 

and 

/,(x)  =  (.i-— a  ){x— a.2){x— a3) -\-{r—ai)(r  —  a2){x—ai) . . .  +,  <fcc, 

it  follows  that 

m± . . . .  _i_+^_+_i_+-j_  ....<„ 

f(x)  x— a4  '  x— a3  '  x— a2  '  x — ax  ' 

In  like  manner,  for  any  other  equation  F(x)  =  0,  we  have 

F(x)  x—b^x—b^x—b^x—bi  .    v  ' 

Suppose  the  two  equations 

/(x)=0,  F(x)  =  0, 
have  a  root  in  common,  viz.,  a^b^  then,  dividing  (1)  by  (2),  we  have 

1  _J_       _1_  1 

/,(.;•)     F(x)  x— ^~*~.t— g^~x— a^x— al 


F,(x)  '  f(x)~  1      ,       1 


-+7-T+ 


x — fc4  '  x — />;,'x  — 1>>~  x — bx 
Hence,  multiplying  numerator  and  denominator  of  the  second  member  by 
r— au  and  then  substituting  for  x  its  value  x=rc/,.  we  have 

/i(«i)    F(».) 

/.(a.)      /(«.), 

'  '  Fi(a,)~F((ll) ' 

from  which  wo  learn,  that  if  any  two  equations  have  a  common  root  a,  and 

their  derived  equations  be  taken,  the  ratio  <>!'  tin   original  polynomials,  when  '.' 

us  put  for  r,  will  l>o  equal  to  the  ratio  of  the  derived  polynomials  when  ,i  it  put 

for  t. 

This  property  furnishes  us  with  a  ready  method  of  determining  the  value 


THEORY  OF  VANISHING  FRACTIONS.  391 

fix) 

of  a  fraction,  such  as  „.   v,  when  both  numerator  and  denom  nator  vanish  for 
F(.r) 

a  particular  value  of  x,  as,  for  instance,  for  x=a.     For  we  shall  merely  have 

to  replace  the  polynomials  in  numerator  and  denominator  by  their  derived 

polynomials,  and  then  make  the  substitution  of  a  for  x.     If,  however,  the 

terms  of  the  new  fraction  should  also  vanish  for  this  value  of  x,  we  must  treat 

it  as  we  did  the  original,  and  so  on,  till  we  arrive  at  a  fraction  of  which  the 

terms  do  not  vanish  for  the  proposed  value  of  .r.     The  following  examples  wfl1 

nufficiently  illustrate  this  method  : 

(1)  Required  the  value  of 

x2— a* 
x — a  ' 
vhen  x=a. 

Ma)      2« 
Here  „  .   ,=-— =2a,  the  required  value 
Fx{a)      1  7 

(2)  Required  the  value  of 

nx^1  —  (n-\-l)xa+l 

(T~rf  '* 

when  .t=1. 

fi(x)      n(n-\-l)xn— n(n+l)xn~l 

¥Jx)=  -2(1  —a:)  • 

This  still  becomes  -  for  x=l, 

fi.(x)      n*(n+l)x*-1— n(»+l)(n— l)s—« 
F^j-  2 

/2(1)      n(n+l) 


the  value  sought. 


F.(l)' 


o 


(3)  Required  the  value  of 


when  .r=l. 


(4)  Required  the  value  of 


1  —  xa 


Ml)      -n 
F^l)--!-"' 


b(a —  \fax) 

a — x      ' 
for  x=a. 

We  may  here  put  \/x=y,  and  thus  change  the  fraction  into 

Ha— a2y) 
a—y* 

My)     -bJ       f^ah     h- 

WhA=~^r  •'•  — ~=o' the  value  reqmred- 


* 


This  is  the  expression  for  the  sum  of  n  terms  of  the  series 


392  ALGEBKA. 

(5)  Required  the  value  of 

m  m 

f(y)    (a+g)°—  (g+y)~ 

F(y)~"  x-y 

when  x--=y. 

Put  a-\-y=zzn,  then  the  fraction  is  chauged  into 

in 

(a-f-.r)n — :m 
x— z"-\-a 

m 

/,(;)      —  mzm~l      m    zm     m     {a-\-y)" 
Fi(z) —  —  n:"-1  ~n  '  za  ~  n  '     a-\-y    ' 
and,  therefore,  the  value,  when  x=y,  is 

m 

m    (a-\-x)a 
n  '     a-\-x 


ELIMINATION. 

RESOLUTION  OF  EQUATIONS  CONT  A I  NINO  TWO  OR  MORE  UNKNOWN 
QUANTITIES  OP  ANY  DEGREE  WHATEVER. 

315.  We  have  already  indicated,  at  p.  157,  the  possibility  of  eliminating  one 
of  two  unknown  quantities  from  two  equations  by  the  method  of  the  common 
divisor.  The  general  theory  of  equations  which  has  since  been  unfolded  will 
afford  the  means  of  giving  a  more  full  development  to  this  subject. 

The  two  given  equations  may  be  thus  expressed  : 

.F(.r,  y)=0,f(x,y)=0 (1) 

They  are  said  to  be  compatible  if  they  have  common  values  of  x  and  y.  This 
is  the  case  with  two  equations  derived  from  the  same  problem,  the  conditions 
of  which,  for  the  determination  of  the  required  quantities,  are  expressed  by 
the  two  given  equations. 

Suppose  now  that  one  of  the  common  values  of  y  were  known,  and  substi- 
tuted for  y  in  the  two  equations  (1),  the  first  members  of  both  would  become 
functions  of  X,  and  known  quantities  ;  the  common  value  of  .r,  corresponding  to 
this  value  of  y,  must  have  tho  property  of  every  root  of  an  equation  pointed 
out  at  Prop.  II.  of  Art.  238  ;  that  is  to  say,  if  a  denote  this  value  of  x,  each 
of  the  equations  (1)  must  be  divisible  by  (x — o)  ;  in  other  words,  they  must 
have  a  common  divisor  containing  x.  If,  therefore,  without  knowing  and  sub- 
stituting the  value  of?/,  wo  proceed  with  the  two  given  equations  (1),  accord- 
ing to  the  method  for  finding  tho  greatest  common  divisor,  until  wo  arrive  at  a 
divisor  of  the  first  degreo  with  respect  to  r,  and  to  a  remainder  independent 
of  .r,  or  containing  only  y,  as  this  remainder  would  have  been  zero  if  I  lie  value 
of  y  had  occupied  its  place  during  ilio  process,  the  value  of  y  ought  to  !>,•  such 
as  to  reduce  this  remainder  to  zero.  The  values  of  y  which  will  do  this  are 
found  by  putting  this  last  remainder  equal  to  zero,  and  thus  forming  what 
called  the  final  equation  in  y  only.     The  values  of  y  which  satisfy  the  final 

equation  are  the  only  compatible  values  of  this  unknown  in  the  two  given  equa- 
tions (l).  The  corresponding  values  of.;-  are  found  by  substituting  these 
vulucs  uf  ?/  successively  in  the  last  divisor,  which  will  ordinarily  he  of  the  flat 


ELIMINATION.  393 

degree  with  respect  to  x,  and  setting  this  equal  to  zero  ;  each  value  of  y  gives, 
!>y  means  of  this  divisor,  the  corresponding  value  of  x,  which,  substituted  with 
.1  in  the  given  equations,  will  satisfy  them.  Should  this  divisor  reduce  to  zero 
by  the  substitution  of  the  value  of  y,  we  must  go  back  to  the  previous  one  of 
the  second  degree,  which,  put  equal  to  zero,  will  furnish  two  values  of  x  for 
each  of  y ;  if  this  reduco  to  0,  we  must  go  to  that  of  the  3°  degree,  and  so  on. 

316.  This  conclusion  may  be  arrived  at  in  another  manner.  Denoting  by 
A=0,  for  simplicity,  the  first  of  the  two  given  equations  F(.r,  y)=0,  and  by 
13  =  0  the  second /(.r,  y)  =  0,  by  Q  the  quotient  of  A  by  B,  and  by  R  the  re- 
mainder, we  have 

A=BQ+R (2) 

It  follows  from  this  equality  that  all  the  values  of  the  unknewn  quantities  x 
and  y,  which  give  A=0  and  B  =  0,  must  also  give  R=0,  since  the  quotient 
Q  can  not  become  infinite  for  finite  values  of  x  and  y,  the  given  equations  be- 
ing supposed  to  be  entire  functions,  or  capable  of  being  rendered  such  with 
respect  to  x  and  y.     (See  Art.  275,  Cor.  2.) 

For  the  same  reason,  all  the  values  which  will  give  B  =  0  and  R=0,  will 
also  give  A=0.  The  system  of  equations  A=0,  B  =  0  may,  therefore,  be 
replaced  by  the  more  simple  system  B=0,  R=0. 

If  now  B  be  divided  by  R,  and  a  new  remainder,  R',  be  reached,  it  may  be 
shown  in  a  similar  manner  that  the  system  B=0,  R=0  can  bo  replaced  by 
the  system  R  =  0,  R'  =  0,  R'  being  of  a  lower  degree  with  respect  to  x  than 
R,  and  so  on,  till  we  arrive  at  a  remainder  independent  of  .r.  Let  R"  be  this 
remainder.  Then  the  original  equations  are  replaced  by  the  system  R'=0, 
R"=0,  in  which  R"=0  is  the  final  equation  in  y  only,  and  R'  generally  of 
the  1°  degree  with  respect  to  .r. 

317.  The  same  conclusion  could  not  have  been  arrived  at  had  y  been  sup- 
posed to  enter  into  any  of  the  denominators  in  the  above  process.  Suppose, 
for  instance,  that  Q  in  equation  (2)  contained  denominators  functions  of  y, 
then  Q,  might  possibly  become  infinite  by  the  values  of  y  reducing  these  de- 
nominators to  zero,  and  BQ  thus  might  bo  finite  (see  Art.  156,.  3°),  though  B 
were  zero. 

318.  If,  in  order  to  prevent  the  occurrence  of  y  in  the  denominator  of  the 
quotient  when  affecting  the  division  of  A  by  B,  it  had  been  necessary  to  mul- 
tiply the  polynomial  A  by  some  function  of  y,  foreign  roots  might  thus  be  in- 
troduced, not  belonging  to  the  proposed  equation.  For,  call  c  this  function, 
and  represent  by  Q  still  the  quotient  obtained  after  this  preparation,  and  by  R 
the  remainder,  we  shall  have 

cA=BQ+R. 

This  equality  proves  that  the  solutions  of  the  equations  B  =  0,  R=0  are  the 
same  as  those  of  the  equations  cA=0,  B=0.  But  this  last  system  divides 
itself  into  two  others,  A  =  0,  B  =  0,  and  c=0,  B  =  0,  consequently  the  equa- 
tions B=0,  R  =  0  will  admit  all  the  solutions  of  the  proposed  equations  ;  but 
they  will  admit,  also,  all  those  of  the  equations  c=0,  B=0,  which  can  not  be- 
long to  the  equation  A  =  0.  The  same  may  be  shown  for  any  foreign  factor 
Decessary  to  bo  introduced  to  effect  any  subsequent  division. 

On  the  other  hand,  factors  are  sometimes  suppressed  for  convenience  in  the 
process  for  finding  the  common  divisor.  If  these  factors  were  such  as  would 
reduce  to  zero  on  attributing  to  y  its  proper  values,  the  process  ous;ht  to  ter 


394  ALGEBRA. 

rmnato,  since  the  whole  remainder  becomes  zero  with  one  of  its  factors,  and 
the  preceding  divisor  would  be  a  common  measure  of  the  two  polynomials; 
and  yet  these  values  of  y  which  produce  this  common  measure  would  not 
have  been  presented  by  tho  final  equation  arrived  at  had  the  factor  in  question 
been  suppressed  without  notice. 

From  tho  foregoing  considerations  we  see  that,  to  obtain  tho  values  of  y 
which  belong  to  the  proposed  equations,  we  must  equate  to  zero  the  remain- 
der which  is  independent  of  x,  as  also  each  of  the  factors  in  y  which  have 
been  suppressed  in  tho  course  of  the  operation,  and  resolve  each  equation 
separately  ;  secondly,  that  among  the  values  thus  obtained  there  maybe  some 
which,  on  trial  in  the  proposed  equations,  prove  extraneous,  and  which  must, 
therefore,  bo  rejected. 

319.  Simplifications  may  sometimes  be  employed,  the  nature  of  which  is 
explained  conveniently  by  the  aid  of  symbols,  as  follows  :  Let  the  polynomials 
A  and  B,  the  first  members  of  the  given  equations,  be  put  under  the  form 

A=dd'd"uu'u",  B=dd'd"vv'v", 
In  which  d  represents  a  common  divisor  of  A  and  B,  containing  x  only  ;  d' 
another,  containing  y  only ;  and  d"  a  third,  containing  both  x  and  y.  The 
other  factors,  w,  u',  u",  v,  v\  v",  have  a  similar  meaning,  except  that  they  are 
not  common  to  the  two  polynomials  A  and  B.  The  proposed  equations  may 
be  satisfied  by  placing  d=0  ;  this  equation  contains  only  x,  and,  when  re- 
solved, furnishes  a  limited  number  of  values  of  this  unknown  quantity,  to 
which  may  be  joined  any  value  whatever  of  y,  and  tho  given  equations  A=0 
and  B  =  0  will  be  satisfied.  Again,  d'  =  Q  will  satisfy  them,  which  gives  simi 
larly  limited  values  for  y,  unlimited  for  x.  Finally,  suppose  d"=0  ;  as  d" 
contains  both  x  and  y,  an  arbitrary  value  may  be  given  to  one  of  the  unknown 
quantities,  and  this  equation  will  make  known  a  corresponding  one  for  the 
other. 

The  other  modes  of  satisfying  tho  given  equations  consist  in  equating  to 
zero  simultaneously  one  of  the  factors  u,  u',  u"  of  the  first,  and  one  v,  v\  or 
v",  of  tho  otheY.  But  v  and  u  can  not  be  simultaneously  equal  to  zero,  since 
they  each  contain  only  .r,  and  are  supposed  to  havo  no  common  divisor,  d  having 
been  understood  to  comprise  all  the  common  factors  depending  on  x  alone. 
For  a  similar  reason,  u'  and  v'  functions  of  y  alone  can  not  at  the  same  timo  be 
equal  to  zero.  But  u"  and  v",  being  put  equal  to  zero,  are  to  be  proceeded 
with  by  the  method  of  tho  common  divisor,  as  already  explained,  and  will  fur- 
nish a  limited  number  of  values  for  y,  and  corresponding  values  limited  also 
for  x. 

320.  Should  the  remainder,  in  seeking  for  a  common  divisor,  not  contain  y, 
but  only  known  quantities,  it  could  not  be  put  equal  to  zero.  In  this  case  tho 
given  equations  would  be  incompatible. 

EXAMPLES. 

(1)  Let  the  equations  bo 

(_2x*+2)y8+(x«— 2a?— 2x»-f  te-f-l^+fx6— 2i8+x)y=s0, 

( _.r-J-  l)j/4-  ( —.«.-'  +  ./■)  y '+  (.r3— x2)y-f  (x*— *»)y«=0. 
There  arc  numerous  simplifications  of  these,  for  they  can  be  decomposed  into 
factors  like  the  following: 

7/(.r-l )(,-+?/)  X  (X+  1  H  I  '-•-V/-1)  =  0, 

y(x-l)(x+y)Xy(*«-3  1=0. 


ELIMINATION.  J95 

Equating  to  zero  first  the  common  factors,  e;ich  in  its  turn,  we  obtain 
'   <  y=0,  <  y  indet,  $  y  indet., 

I  x  indet.,  \  x=l,  \  x  =  — y  ; 

next  equating  to  zero  the  other  factors,  we  have  four  systems  of  equations,  viz. 

First  system  )  ^~  S  whence   \  ^=      , 

J  lx-\-l=U    $  I  x=  —  1. 

Second  system   <  -{~  n       ,      „    >   whence   <^      ,    )  2/  =      , 
J  ^  x- — 2y — 1=0   $  }  x=l,  \x—  —  1. 

Third  system  j  *  "f =Jj  |  whence  \  ^-l  j  ^ 

Fourth  system  5  ^"^  =  °        l  whence  \V  =  1  +  V2    _  (  y=l+  V2   _ 
J  a*— 2y-,l=0  S  \  x=±(l  +  V2)  J  *s=±(l—  \/2) 

In  the  first  three  systems,  all  the  solutions,  except  .r= — 1,  y= — 1,  have 
already  been  found;  in  the  fourth,  those  in  which  we  have  x= — y  are  also 
already  known ;  hence,  in  reality,  wo  have  only  determined  three  new  solu- 
tions, viz., 

C7/=_l      cyssi+va      cy=i-Vj 

\x=-\,      \x=l+y/2,      lx=l-V*- 
(2)  To  resolve  the  two  equations  0 

x3— 3yx2-|-(3y2— 3y+l)x— y3-f-y2— 2y  =  0, 
&— 2yx  +y»_ y=0. 
These  equations  can  not  be  decomposed  into  factors ;  hence  wo  pass  imme- 
diately to  successive  d 'visions.    This  remark  will  apply  also  to  equations  3  and  4. 

First  Division. 


x3-3ya?+{3y»-y4-l)x-y8+y3-2y 

-i-x'— 2y — x-{  —7/  +  / )  /• 


x3 — 2yx-\-y- — y 


x—y 


-  y*9+(2ys+l)a:-y3-T-ys--2y 

—  yxi-\-2y-x — y*-{-  y" 

x— 2y 

Second  Division, 
x- — 2yx-\-y- — y\x — 2y 
-4-x2 — 2yx  \x 

¥^- 

Hence,  the  final  equations  are  x — 2y=0,  y" — y=0.     "We  deduce  from 
these 

and  as  we  have  neither  introduced  nor  suppressed  any  factor,  these  •  wo  solu- 
tions are  those  of  the  proposed  equations  themselves. 
(3)  To  resolve  the  two  equations, 

(y— l)a«+*ac— 5y+3=0, 

yx°+9x  —  l0y=0. 


First  Division. 


(y—l)  x9-f  2x   — 5j/+3 
(y— lJya^+Syx— 5y=+3y 


yx*-\-9x—l0y 


y-i 

-f(y  —  l)yx"  —  (—  9y-f-  9)x  —  10y"-+10y 
(_7y+9)*+  5y»-  7y. 


30C  ALGEBRA. 

As  we  have  multiplied  by  y,  it  is  necessary  to  resolve  the  equations  y=0, 
yx -'  +  9x  — lOy =  0,  which  give  x=0,  y=0,  and  to  examine  whether  these 
values  make  the  dividend  equal  to  zero.  As  this  is  not  the  case,  it  follows  that 
they  form  a  foreign  solution,  which  it  will  be  necessary  to  suppress. 


Second  Division. 


{-7y+9)*+fy—7y 


yx  +  (_5i/'  +  7»r-63y  +  81) 


yx2  +  9x— lOy 
(-7y  +  9)yx"-+(-G3y+81)x+70if-00y 

( ~ 7y + 9)yx' -  ( -  57/3+  7yy.x 

(_5i/>  +  7y-  —  6:Jy  +  81).<-+70y  —  90y 

(—5,f+7f— 63y+81)(— 7y+9)z-490^+1200?/2_ 81Qy 
( -5^+7^— 63.y+81)(—7y+9)3.-—257y5+70y— 364^+8463/-— 5G7,y 

25?/s— 70?/4— 126.y3+414?y3— 243w. 

The  equations  which  it  is  neccssaiy  to  resolve  are 

(_73/+9).r+5y2— 7?/  =  0, 

25y5— 70y4— 120y:!4-414?/2  — 243y  =  0. 

The  second  gives  the  results,  which  may  be  readily  verified, 

— 3rb3V"I6 
2/=0,  yrsl,  y=3,  y= ^ . 

By  substituting  these  values  in  the  first  of  the  given  equations,  we  obtain  for 
x  the  corresponding  values  x=0,  1=1,  ar=2,  rr= — 5+  -v/10. 

In  the  second  division  we  have  been  compelled  to  multiply  by  —  7y-\-9,  but 
no  foreign  solution  has  been  introduced. 

We  have,  then,  only  to  suppress,  in  the  five  solutions  above,  that  which  has 
been  introduced  by  the  first  division.     There  remain,  then,  for  the  given  equi 
tions  the  four  following  solutions  : 

(         _3_l_3-v/l0  r         —3— 3-/10 

ix—l,         lx—2t         (x=_5_A/io,  lx=—  5+-/10. 

(4)  Let  the  equations  be 

z2+(8i/-13).r-f7/-73/+12=0, 
a*_(4y+  l).r+2/3+5?y=0. 

.Firs£  Division. 
z*  +  (dy-l3)x+if—7y  +  Uh*-(iy  +  l)x+y°-+5y 
+x2-(4y+  l)x+y'+5y         |l 
(12y—  12).r— 12y-f-12 
This  remainder  can  be  decomposed  into  tho  factors  12(y  —  l)(x  —  1);  the 
calculations  will  be  simplified,  and  wo  shall  have  those  two  systems  of  equa- 
tions : 

y  — 1  =  0  (r— 1=0 

is-(4y+l)z+y9+5y=0,  i  i»-(4y+l)x- 
Each  of  these  can  bo  at  once  resolved,  and  we  find 


\ 


Cy  =  l   Cy=l   <7/  =  o  S.'/=—  1 
(x=2,(x=2,  ).r=l.  <>  tsl, 
(5)  r,-r-2y.r-+2y(?/— 2)r4-v:— 4  =  0. 

z"+2yx+2y»— 5y+2=0.  fys=2      JyssS 


Ans. 


METHOD  OF  LAIJATIE  397 

(6)  z*— 3yx*+3x2+3y2x— 6yx— x— y+3t/2+2/~  3=°. 
r3-}- 3?/:r2— 3.c24-3?/2i-— 6yx— X+y3— 3?/2— i/+3  =  0. 

I1  irst  system      <  ^  <  ^  <  •/ 

^  <a-=0,  4  x=2,  lx=—  2. 

Second  system  \  *  )  ^  <|  %         \  y  —  ~ 

(  x=l,  (  x=  —  1,  I  x=l,  I  x= — 1. 

Third  system    \V  =  3  SV=  — 1 
(x=0,(x=0. 

(7)  z*+yx*-(f~+l)x+y-if=0, 
a.-3_2/x2_(7/2+G?/+9).r+2/3+G^2+93/=:0. 

The  first  division  gives  the  remainder 

2/^+(37+4).r-(2/3+37/2+47/). 

To  be  able  to  perform  the  second  division,  we  multiply  the  dividend  by  y , 
in  the  same  way  we  prepare  the  first  remainder  to  be  divided.  We  thus  ar- 
rive at  a  remainder  of  the  first  degree  in  x,  which  can  be  put  under  the  form 

32(y*+3y+2)(x-y). 

Dividing,  then,  the  remainder  of  the  second  degree  by  x — y,  we  obtain  the 
quotient 

yx+if+3y +  4  =  0, 
and  there  is  no  remainder. 

From  these  calculations  we  conclude  that  the  first  members  of  the  proposed 
equations  are  divisible  by  x — y,  so  that  they  can  be  verified  by  all  the  solutions 
of  the  indeterminate  equation  x — ?/=0.  The  other  solutions  are  furnished 
by  the  system  of  two  equations, 

if+3y  +  2=0,  yx+7f+3y+4=0; 
hence  we  obtain  tl  e  solutions 

y=-l,x=  +  2;  y=-2,  r  =  +  l. 

METHOD  OF  LABATIE. 

321.  Having  thus  stated  the  principles  on  which  the  ordinary  method  of 
elimination  depends,  we  shall  now  proceed  to  show  how  this  method  has 
lately  been  perfected  by  Labatie  and  Sarrus.  By  the  aid  of  the  theory  which 
they  have  introduced,  we  shall  be  able  to  perform  the  required  eliminations 
without  introducing  any  foreign  solutions. 

Suppose  that  A  and  B  represent  the  quotients  which  we  obtain  by  dividing 
the  first  members  of  the  given  equations  by  all  of  their  factors  which  depend 
only  on  y. 

Let  c  be  the  factor  by  which  it  is  necessary  to  multiply  A,  in  order  that  we 
may  be  able  to  divide  it  by  B  ;  represent  by  q  the  quotient  that  we  obtain  in 
this  division,  and  by  Rr  the  remainder,  r  designating  those  factors  of  this  re- 
mainder that  are  not  dependent  on  x.  Let  ct  be-  the  factor  by  which  we 
must  multiply  B  to  render  it  divisible  by  R  ;  represent  by  qx  the  quotient,  and 
by  "RiTi  the  quotient  that  we  obtain  in  this  second  division,  ?-j  designating  the 
product  of  those  factors  of  this  remainder  which  do  not  depend  on  x,  and  so 
on.  Finally,  suppose,  for  the  sake  of  simplicity,  that  at  the  fourth  division 
we  obtain  a  remainder  independent  of  x,  and  designate  this  remainder  by  r3. 
We  have  the  equalities 


"J93  ALGEBRA, 

c A  =B  q +  Rr 

CiB   =  R  qi-\-\\lri /jx 

c_.Il  =  K;y  i-f-RsTj 
(_c;iR1  =  i;.7  .+r3 

Let  J  be  the  greatest  common  divisor  of  c  and  r,  rf_  the  greatest  common 

CC_  CC1C9  CC^r.j 

divisor  of —  and  r,,  </>  th:it  of  -ry  and  r2,  d3  that  of   ,  ,",-  and  r3.     We  shall 

proceed  to  prove  that  we  can  obtain  all  the  solutions  of  the  system  A  =  0, 
B  =  0,  without  any  foreign  solution,  by  resolving  the  following  systems  : 

(  r  i  r{  i  r.  (  rn 

\d~  \dx~  j  dz  j  d3  (2) 

(b=o,       (r=o,      (ri=o,      (r,=o 

To  establish  this  proposition,  we  shall  first  prove  that  the  solutions  of  the 
systems  (2)  all  agree  with  those  of  the  system  A  =  0,  B=0;  we  shall  after- 
ward show  that  the  solutions  of  the  system  A  =  0,  B  =  0,  are  all  comprised  in 
those  of  the  systems  (2). 

[a]  Dividing  by  d  the  two  members  of  the  first  equation  of  system  (1),  it 
becomes 

3A4B+5R <3> 

■4  is  entire,  for  c  andr,  by  hypothesis,  are  divisible  by  d  ;  hence,  qB  is  divisible 

by  d ;  but  B,  by  hypothesis,  is  prime  with  respect  to  d  ;  therefore,  d  divides  q. 

Equation  (3)  shows  that  the  values  of  x  and  y,  which  satisfy  the  equations 

r  c  c  r 

B  =  0,  "j  =  0,  destroy  also  jA.;  but  -3  and  -  are  prime  with  respect  to  each 

other.     Consequently,  1°,  all  the  solutions  of  the  system  B  =  0,  -j  =  0,  agree  tcith 

those  of  the  system  A=0,  B=0. 

r_ 
[h]  To  obtain  a  relation  between  A,  R,  and  -y,  we  multiply  equation  (3)  by 

c.,  and  in  the  resulting  equations  place,  instead  of  c.B,  its  value  as  found  in  the 
second  member  of  the  second  equation  of  system  (1) ;  wo  thus  obtain 


-dA=\— -d~)R+driRl- 


The  quantity  - — -, — -  is  entire,  because  r  and  q  are  divisible  by  J :  more- 

over,  tliis  quantity  is  divisible  by  dv ;  for  dv  divides  -j  and  r„  and  it  is  prime 

with  respect  to  R.     Dividing  the  two  members  of  the  above  equation  by  du 

q  C\T-\-qq\ 

and  taking,  to  abridge,  *5=M,  — -p —  =  Mi,  it  becomes 

^-A  =  M1R+MR,y     (4) 

dd\  di 

To  obtain  a  relation  between  B,  R,  and  y,  wo  first  multiply  the  second 

c  fcC]         cqx  c  <v, 

equation  of  system  (1)  by  -.,  which  gives  -yB*=-yl\4--fRi>V     bince  -r  and 


METHOD  OF  LABAT1E.  399 

rx  are,  by  hypothesis,  divisible  by  du  it  follows  that  dx  divides  also  -jR  ;   but 

dx  is  prime  with  respect  to  R ;  hence,  dx  divides  —r.     Div  iding  all  the  terms  of 

c  cq\ 

the  equation  by  du  and  taking,  to  abridge,  -y=N,  -jy  =N;,  it  becomes 

^B=N,R+NR,^ (5) 

Equations  (4)  and  (5)  prove  that  all  the  values  of  x  and  y,  which  reduce  the 

rx  CC\  cci  cci  r{ 

polynomials  R  and  -r  to  zero,  destroy  also  -jr A  and  -3-7-B  i  DUt  ~tt  and  -r 

are  prime  with  respect  to  each  other;  consequently,  2°,  all  the  solutions  of 
the  system  R=0,  -j-  =  0,  agree  with  those  of  the  given  system,  A  =  0,  B  =  0. 

[c]  We  obtain  a  relation  between  A,  Rn  and  -j-,  by  multiplying  equation  (4) 

by  c2,  and  placing,  instead  of  c2R,  its  value  found  in  the  second  member  of  tho 
third  equation  of  system  (1) ;  we  thus  find 

By  hypothesis,  d3  divides  the  first  member  of  this  equation,  it  also  divides  r3 ; 
it  ought,  then,  to  divide  RJ  Mj q.2-\-  Mc^-r-j;  but  Rt  and  d2  are  prime  with  re- 
spect to  each  other ;  d%  then  divides  the  term  by  which  Rt  in  the  above  equa- 
tion is  multiplied.     Designating  the  quotient  by  M2  the  equation  becomes 

3HA-"*+**a <6> 

Multiplying  equation  (5)  by  c2,  and  then  placing,  instead  of  c2R,  its  value 
found  in  the  second  member  of  the  third  equation  of  system  (1),  it  becomes 

^B=RI(N1,:+No^)+N1R,r, 

We  can  demonstrate  as  before  that  the  multiplier  of  Ri  is  divisible  by  d3, 
and,  representing  the  quotient  by  N2,  we  find 

•    SIB=N-R'+N'E^ •  •  •  •  •  (7> 

Equations  (6)  and  (7)  show  that  all  the  values  of  x  and  y,  which  reduce  the 

r2 
polynomials  Rt  and  -f  to  zero,  destroy  also  the  first  members  of  these  two 

«2 

cciCj        ,  r2  ...  ,       , 

equations ;  but    ,  ,  ;   and  -j  are  prime  with  respect  to  each  other ;  conse- 

quently,  3°,  all  the  solutions  of  the  system  R!=0,  ■y-=0,  s"^  ^l0se  of  the  pro 

posed  system,  A=0,  B=0. 

r3 

[d]  The  equation  which  gives  a  relation  between  A,  R2,  and  ~r,  can  be  ob 

tnined  by  multiplying  equation  (6)  by  c3,  and  placing,  instead  of  c3Rj,  its  value 
as  i^iven  in  the  second  member  of  the  fourth  equation  of  system  (1)  ;  we  thus 
Snd 


400  ALGEBRA. 

^A=R;(.r7;;+(,M^)+M,r, 
Dividing  the  two  members  of  this  equation  by  da,  and  designating  by  M3  the  quo 
tient  obtained  by  dividing  the  entire  polynomial  M2</3+c3Mi-t-  oy  d^.  theie 
results 

To  obtain  a  relation  between  B,  R:,  and  y,  wo  multiply  equation  (7)  !•• 

and  put  in  iho  place  of  c3Ki  the  second  member  of  the  fourth  equation  of  the 
system  (1),  which  gives 

cCxCjPa 


gB=R2(N,73+C3N^)+N2r3. 


dd-ji  a 
Dividing  both  members  by  dz,  and  designating  by  N3  the  quotient  obtained 

by  dividing  the  entire  polynomial  N^a-J-^Ni-T-  by  ds,  it  becomes 

Jg|B=N,K,+N>2 CD 

Equations  (8)  and  (9)  show  that  all  the  values  of  x  and  y,  which  reduce  the 

r3 
polynomials  R2  and  -j-  to  zero,  destroy  also  the  first  members  of  those  equa- 
ls 

tions  ;  but   , ,  ,  ,  and  —  are  prune  with  respect  to  each  other  ;  consequent 

r3 
ly,  4°,  all  the  solutions  of  the  system  R2=0,  ^-=0   concur   with   those  of  the 

•proposed  system,  A  =  0,  B=0. 

(II.)  It  remains  still  to  be  proved  that  any  system  whatsoever  ofwalucs 
which  satisfy  the  equations  A  =  0,  B  =  0,  is  a  part  of  the  systems  of  values 
which  furnish  equations  (-2).  • 

To  form  the  equations  which  demonstrate  this  second  part  of  the  theorem, 

c  q 

let  us  first  place  in  equation  (3)  N  instead  of  -3,  and  M  instead  of  -% ;  it  will 

become,  transposing  the  term  MB, 

NA  —  MB  =  R^ (10)  % 

Eliminate  now  R  between  equations  (4)  and  (5).  Wo  can  oflfect  this  elim- 
ination by  subtracting  one  of  these  equations  from  the  other,  alter  wo  have 
multiplied  tho  first  by  Ni,  the  second  by  M  .  remembering  the  values  previ 
ously  given  to  Ni  and  ML ;  but  the  calculations  will  be  simpler  if  we  multiply 
equation  (4)  by  B  and  equation  (5)  by  A.  Subtracting  the  two  rosidting  equa- 
tions the  one  from  tho  other,  we  find 

(MtB— N,  ^)R+(MB— NA)R,~0. 

Placing  instead  of  MB  —  NA  its  \a!ue  previously  determined,  — R-j.  and 
suppressing  tho  factor  R,.  this  equation  becomes 

N,A  — M,l!  =  —  R,-r/   ....   (11) 


METHOD  OF  LABATIE.  401 

Finally,  we  eliminate  Rx  between  equations  (G)  and  (7).     To  do  this,  mul 
tiply  equation  (6)  by  B  and  equation  (7)  by  A ;  then  subtract  the  one  of  the 
resulting  equations  from  the  other,  wo  thus  obtain 

(M3— NsAJRi+CMjB— NiA.)Rsft=0, 

Placing  in  this  equation,  instead  of  MiB— NXA,  its  value,  determined  in  (11  j, 
Rj-tt,  and  suppressing  the  factor  RH  it  becomes 

■         Hi_I(fl-fcS£ (12) 

In  the  same  manner  we  obtain  the  equation 

Equation  13  shows  that  eveiy  system  of  values  of  x  and  y  which  gives 
A=0,  B=0,  ought  also  to  satisfy  the  equation 

~Lr-i.r±rjL—c\ 
d  d\d>ids 

an  equation  which  requires  that  one  of  its  factors  equal  zero,  whence  it  fol 
lows  that  the  equations 

r         rx  r2  r3 

3=°-  ir"'  5=0'  3-3=°' 

give  all  the  correct  values  of  y. 

This  being  established,  let  x=a,  y=z(i  be  a  system  of  correct  values  of  the 

equations  A  =  0,  B=0. 

r 
Jf  the  value  y=P  is  a  root  of  the  equation  -^=0,  it  is  clear  that  the  system 

r 
r=a,  2/=/3  will  be  a  solution  of  the  system  B  =  0,  -i=0. 

r 
If  the  value  y=P  does  not  verify  the  equation  -y=0,  and  if  it  is  a  root  of 

the  equation  -f=0,  we  perceive,  by  equation  (10),  that  the  system  x=a, 

dl 

y=fi  will  give  R=0  ;  consequently,  it  will  be  a  solution  of  the  system  R=0 

r  Ti 

Tf  the  value  y=fi  verifies  neither  the  equation  -7=0  nor  the  equation  ^-=0, 

and  is  a  root  of  the  equation  -f=0,  we  see,  by  equation  (11),  that  the  system 

d% 

r=a,  y=P  will  give  Ri=0  ;  consequently,  it  will  be  a  solution  of  the  system 

R1==0,  5=0. 

r  Ti 

If  the  value  y=(3  does  not  verify  any  one  of  the  equations  ^=0,  ^"=°> 

y  =  0,  and  is  a  root  of  the  equation  -r=0,  we  see  by  equation  (12)  that  the 
aystem  x=a,  y=@,  will  give  R3=0 ;  conseruently,  it  will  be  a  solution  of  the 

ystem  R;=0,  "T=0. 

C  c 


402  ALGEBRA. 

Hence,  all  the  systems  of  values  which  satisfy  tiie  equations  A  =  0,  B^eO, 
form  part  of  the  values  which  furnisli  equations  ("-')• 

The  equation  ~j-~r  --J  •_r  =  0,  which  gives  all  the  correct  values  oi  y,  it 
called  the  final  equation  in  y. 

REMARKS  ON  THE   PRECEDING    METHOD. 

y 

It  may  chance  that  in  one  of  the  equations  of  system  (2),  for  example,  y 

«i 

=0,  R  =  0,  a  value  of  y,  derived  from  the  first  equation,  destroys  some  of  the 
coefficients  of  the  powers  of  r  in  the  second  equation,  after  the  highest  power 
of  x ;  in  this  caso  wo  only  obtain  a  Dumber  of  #Iues  of  x  inferior  to  the  de- 
gree of  the  equation  11=0;  and  if  the  substitution  of  the  value  of 
destroy  all  the  multipliers  of  tho  powers  of  x  in  R,  the  equation  R=0  v 
not  give  any  value  of  x.     In  fact,  it  can  bo  proved,  by  a  method  similar  to  that 
which  wo  have  employed  with  reference  to  the  general  equation  of  the  second 
degree  (Art.  191),  that  if  in  an  equation  of  the  form  Sx"+HxB  '-f-K 
-f-  . . .  =0,  we  suppose  that  tho  quantities  which  enter  into  the  coefficients 
S,  H,  K,  occ,  are  of  such  a  nature  that  wo  have  S  =  0,  H=0,  &c.,  the  equation 
lias  infinite  roots  equal  in  number  to  the  consecutive  coefficients  which  are  re- 
duced to  zero.     But  it  should  bo  remarked  that  the  theory  by  which  we  have 
proved  that  the  solutions  of  systems  (2)  are  the  same  with  those  of  the  system 
A  =  0,  B  =  0,  only  applies  to  solutions  expressed  by  finite  values  of  x  and  >/. 

To  prove  that  the  solutions  of  systems  (2),  in  which  the  value  of  X  is  in- 
finity, also  suit  the  proposed  equations  A=0,  B=0,  suppose  that  y  =   . 

fying  the  equation  —  =  0,  causes  one  or  more  of  the  multipliers  of  the  higher 
powers  of  x  in  R  to  vanish.     If,  in  tho  two  members  of  tho  equality  (4)  we 

T 

make  y=[3,  the  term  MR,j  will  bo  reduced  to  zero,  and  the  degree  of  the 

term  M,R  will  be  lowered  with  respect  to  x  one  or  more  units. 

Again,  we  can  not  suppose  thai  the  terms  of  3ItR  which  are  reduce*]  to 

zero,  havo  been  destroyed,  until  we  have  made  y=/3  in  tho  terms  of  MR,-4, 

because  the  degrees  of  A,  B,  R,  Ri,  &c.,  are  decreasing,  and  we  see  without 
difficulty,  from  the  relations  which  exist  between  M,  Mj,  M;,  &*".,  that  the 
degrees  of  these  quantities  with  respeel  to  x  go  on  increasing.     It  will  be 

necessary,  then,  in  order  thai  y  may  have  the  vain.'  \  thai  the  de>  \ 

with  respect  to  x  be  lowered  as  many  units  a>  the  degree  of  R  is  lowered. 
We  can  prove,  in  tho  same  manner,  thai  the  value  y  =  3  ought  also  to  cause 
one  or  more  of  tho  coefficients  of  tho  higher  powers  t  f  x  in  B  to  vanish.  The 
equations  A  =  0,  B=0  will  give  then  for  '/=  ;  one  or  more  infinite  values  i 

As  to  the  reciprocal  proposition,  thai  the  solutions  of  the  equations  V=0. 
B=0,  in  which  z  is  infinite,  oughl  to  be  found  amon.;  the  solutions  oi  systems 
(2),  it  is  not  the  fact,  as  will  bo  seen  in  the  Becond  example  following. 

i  \  vmi'i.i:  |. 

(y-i)*»+y(y+i)*M-(3y,+2'-2)x+2y= 

(y-l)x»+y(y+l)x  +  3y«-l=G. 


ELIMINATION.  41 3 

The  first  division  gives  at  once  the  remainder  (y —  l)x-\-2y ;  taking  this  re- 
mainder for  a  divisor,  we  obtain,  without  any  preparation,  the  remainder  y"  —  1. 
We  shall  obtain  then  all  tho  solutions  of  the  proposed  system  by  resolving  the 
equations 

3/3—1=0,  (y— l)x+2y=0. 

The  first  equation  gives  j=il.  For  the  value  y=  —  l  we  find  x=  —  1, 
and  this  system  will  satisfy  tho  proposed  equations.  For  the  value  y=-\-l 
wo  find  a.-=oo.  This  system,  also,  will  satisfy  the  proposed  equations;  for 
dividing  each  of  these  equations  by  tho  highest  power  of  X,  and  taking  x=oo, 
tho  two  equations  will  be  reduced  to  y  — 1  =  0. 

EXAMPLE  II. 

(y-l)x"-+yx+y°--2y=0, 
(y-l)XJpy  =  Q. 

The  division  gives  the  remainder  y°~  —  2y  —  0;  the  solutions,  therefore,  of  the 
proposed  equations  depend  on  the  system 

y*-2y=0,  (y-l)x+y  =  0. 
These  equations  give  the  two  systems 

y=0,  x=0;  y  =  2,  x=—2. 
But  the  proposed  equations  possess,  besides,  another  solution,  y=l,  Xs=<x>, 
since  the  value  y=l  causes  the  multiplier  of  tho  highest  power  of  x  in  each 
of  these  equatious  to  vanish. 

322.  The  following  method  of  elimination  avoids  the  introduction  of  foreign 
roots,  and  enables  us  to  determine  tho  degree  of  the  final  equation  : 

Let  equation  A  or  .rm-L.Pxm_1-l-Qcm_2 +T.r-f- V  be  supposed  equal  to 

{x— a)(xm-1  +  Axm-2+B.tm-3+,  &c.)    .  .  .  .  C; 
and  equation  B  or  ^-fP'x^  +  Q'.x"-2  . . .  +T'x+  V  to 

(x— a)(xn-I  +  A'.rn-3+B'a.-n-3+,  &c.)  .  .  .  .  D; 
also,  let  equation  A  be  multiplied  by  xa~l-{- A'.rD_2+B'xn-3,  &c,  and  equation 
B  be  multiplied  by  xm-1-f-A.rm_2-L-B.rm_3,  &c,  it  is  evident  that  the  products 
must  be  equal ;  therefore, 

(zm + P.-cm-1 + Q.rm-2 + ,  &c. ) (.r"-1  +  A'.r"-2 + B'x"~s + ,  &c. )  =  (.in  +  P 'af1  -f 
Q'xn-s^.,  &c.)(xm-1+Axm-2+Bxm-34-,  &c.) E. 

Performing  the  multiplications  and  making  equal  to  each  other,  the  coeffi- 
cients of  the  same  powers  of  x  (Art.  209),  m-\-n— 1  equations  are  obtained 

between  the  indeterminate  quantities  A,  B,  C,  . . . .  A',  B',  C, Now, 

the  number  of  indeterminate  quantities  in  equation  C  is  m — 1,  and  in  equation 
D,  n  —  1  ;  therefore,  the  number  in  equation  E  is  m-\-?i — 2.  Of  the  m-\-n  —  1 
equations  m-\-n — 2  suffice  to  determine  A,  B,  C, . .  .A',  B',  C, . .  . . ;  and  one 

equation  remains  between  P,  Q,  R P',  Q',  R' . .  . .,  which  it  is  necessary 

to  satisfy  in  such  a  manner  that  the  equations  C,  D  may  have  a  common  di- 
visor, x — a;  this  equation  of  condition  is  the  final  equation  required. 

Since  none  of  the  indeterminate  quantities  A,  B,  C  . . .  A',  B',  C  ....  is 
multiplied  by  itself,  the  equations  by  means  of  which  those  quantities  are  de 
termined  are  of  the  first  degree. 

The  final  equation  being  resolved,  and  the  values  of  y  successively  substituted 
in  A.  B,  C, .  . . .  A',  B',  C, .  . .,  the  results  are  obtained  from  the  division  of  the 
polynomials  C,  D  by  the  common  divisor  x  —  a. 


ALGEBRA. 

If  the  equations  A.  B  are  incomplete,  the  two  products  E  can  not  be  com- 
plete polynomials  of  the  degree  rre-J-n— 1;  but  the  terms  which  are  deficient 
io  one  are  found  in  the. other.     For,  taking  the  least  favorable  case,  viz.. 
xm+P=0;  xn+P'  =  0; 
the  identity  which  results  from  the  equality  of  the  two  products  is 

(xm+P)(xn-1+A'xD-*+.  &c.)  =  (xn-fP')(xm-14-Axra--+,  &c.j 

* 

EXAMPLE. 

Let  x2+Px+Q=0; 

X2+P'x+Q'  =  0. 
Denoting  by  x— a  the  factor  which  is  to  bo  rendered  common  to  these  equ& 
.  I  ions  by  the  suitable  determination  of  y,  the  first  equation  may  be  considerea 
the  product  of  x— a  by  a  factor,  x+ A,  of  (tie  first  degree  ;  and  the  second  the 
product  of  x— a  by  a  factor,  x+A',  also  of  the  first  degree. 

...  a-*+p  z+Q  =  (x-a)(x+A), 
i«+P'x+Q'  =  (x— a)(x+A'), 
and  (.r=+P.r+Q)(x+A')  =  (x2+P'x+Q')(x+A), 


a?+P 

+  A' 


x2-|-Q'  x+AQ'. 
+  AP' 


x2_j_Q   x+QA'rzz.r'+P' 

+  PA'  +A 

Making  the  coefficients  of  the  same  powers  of  x  equal  to  each  other, 

P  +  A'=P'  +  A      or         A-A'=P-P' (1) 

Q+PA'  =  Q'  +  AP'orAP'-PA'=Q-Q' (2) 

QA'  =  AQ'  orAQ'  —  QA'  =  0 (3) 

By  mean  of  these  three  equations  of  the  first  degree  the  two  indeterminate 
quantities  A,  A'  can  bo  eliminated,  and  a  single  equation  obtained  in  terms  of 
the  quantities  P,  Q,  P',  Q'- 

For,  if  from  equation  (1),  multiplied  by  P,  or  AP  — PA'  =  (P  — P')P,  equa 
tion  (2)  be  subtracted,  or  AP'  —  PA'  =  Q  —  Q',  the  remainder  is 

AP-AP'  =  (P-P')P-(Q-Q')- 
(P-P')P-(Q-Q') 
Whence  A  = p p,  • 

q.  .,  ,  a,    (P-p,)P,-(Q-Q/) 

Similarly,  A  = p_p, • 

If  these  values  of  A,  A'  are  substituted  in  equation  (3), 

(P-P')P-(Q-Q')v0,     (P-P-)P'-(Q-Q') 
p3p7  XH—  P_P'  xv£_u, 

or         (P_P')PQ'-(Q-Q')Q'-(P-P')Qp'+(Q-Q)Q=0- 
(P_P')(PQ'_QP')+(Q-Q')(Q-Q')=o, 

(P_P')(PQ'-QP')  +  (Q-QT  =  0. 

The  quantities  P,  P',  Q,  Q',  containing  only  y  and  known  quantities,  this  is 
the  final  equation  in  y. 

It  has  been  already  noticed  that,  if  this  equation  is  identical,  the  prop 
equations  have  at  least  ono  common  factor  of  the  form  x— a.  whatever  be  the 
value  of  y  ;  and  that,  if  it  contains  only  known  quantities,  theso  equations  are 
contradictory. 

When  tho  final  equation  has  the  proper  form,  the  factor  r— a  is  obtained  by 
dividing  tho  first  of  the  proposed  equations  by  x-\-A  ;  thus. 


THE  DEGREE  OF  THE  FINAL  EQUATION.  4G5 

x+A)  x-+Px+Q  (.r+P-A 

X2  +  A.T 

(P-A)a:+Q 

(P— A).r+(P  — A)A 

Q— (P-A)A. 

The  quotient  is  x-{-P — A,  and  the  remainder  is  considered  equal  to  zero, 
because  it  is  reduced  to  zero  by  the  substitution,  for  y,  of  a  value  deduced 
from  the  final  equation. 

Making  the  quotient  .r-j-P  —  A  equal  to  zero,  the  value  of  x  is  x=A  — P, 
and  by  substituting  the  value  of  A, 

(P-P')P-(Q-Q') 


or 


P  — P' 

Q-Q' 


-P, 


This  example  is  given  as  an  illustration  of  the  general  method.     From  its 
particular  form  it  admits  of  resolution  by  another  and  a  much  shorter  process. 
For  if  from  x2+Px+Q=0 

x2-|-P'-r+Q'=0  is  subtracted, 
the  remainder  is 

(P_P')x+Q_Q'=0; 

.    .    .(, p p7. 

OP  THE  DEGREE  OP  THE  PINAL  EQUATION. 
323.  The  degree  of  the  final  equation  can  not  be  depressed  by  the  reduction 
of  each  of  the  coefficients  P,  Q,  R . . .  P',  Q',  R' . . .  in  the  equations 

(,n_J_p7.m-l_|_Qrm-2    m    m    t    m     _)_Tx+V=0, 

.r"4-P'2n~1+Q'-'.'n~2  ....  +T'.r+V/  =  0, 
to  the  term  of  the  highest  exponent  in  y  which  it  contains  ;  for  the  degree  of 
each  of  the  equations  is  not  changed  by  the  reduction.    Therefore,  the  reason- 
ing may  bo  applied  to  the  equations 

xm-irayxm~1  -\-lifxm-2 -{-(ym-lx-\-vym=0   ....  (1) 

x"  -fa'2/:cn-I-r-&y.rn-2 +?y*-1x+v'yn  =0  ....  (2) 

which  are  of  the  same  degree  respectively  as  the  preceding  equations.     The 
latter  are  reducible  to 

/x\m        /x\m~l        /.r\m-2  x 

\y)    +a[y)       +h\y)         '  '   '   '   +ty+V  =°     <3> 

\y)+aiy)      +biy)        V..-+^+«'=0     (*) 

x 
in  which  the  unknown  quantity  is  -,  and  a,  b,  . .  .1,  v ;  a',  b', . . .  t',  v',  ar» 

numbers. 

Denoting  by  a,  (3,  y . . .  the  numerictv  roots  of  equation  (3) 
and  by  a',  /3',  y'  .  .  .the  numerical  roots  of  equation  (4) 
these  equations  may  be  decomposed  into 


40G  ALGEBRA. 

Whence    (a; — ay  )(x —  &y){x — yy ),  cVc.  =0 (5) 

(x-ay'){x-!i'y)\X-7'D),  Sec.  =0 (6) 

Substituting  in  equation   (5)   the  roots   of  .r  from   equation  (6),  viz.,  ay 
B'y,  &c., 

(a'y— ay){a'y— 0y){a'y— yy),  &c.  =0, 

ip'y—*yWy—PyWy—7y)*  &c.  =o, 

{y'y  —  ay){y'y—iiy){)'y  —  }y),  &c.  =0. 

Or,  since  the  number  of  factors  in  equation  (5)  is  m,  and  the  number  of 
roots  in  equation  (G)  is  n, 

y^(a'-a){a'-l3)(a'-7),  &C.  =0, 
y™(0' -a)(/3' -0){0'-y),  &c.  =0, 
y-(/_a)(/-j8)(/-7),  &c.  =0. 

Consequently,  there  are  n  equations,  each  of  the  degree  ?»  ;  these  give  all 
the  solutions  in  y.  The  product  of  these  roots  (or  solutions)  of  y  is  the  final 
equation,  sinco  it  becomes  zero  for  all  the  values  of  y  which  render  its  factors 
zero,  and  only  for  these  values.  Now,  this  product  is  evidently  of  the  degree 
tnn.  Consequently,  the  degree  of  the  final  equation  (unless  roots  not  belong- 
ing to  the  proposed  equations  are  introduced  by  the  process  of  elimination) 
can  not  exceed  the  product  of  the  degrees  of  the  proposed  equations. 

It  ought  to  be  observed  that  the  numerical  values  of  the  roots  of  y  are 
changed  by  this  process,  but  that  their  number  remains  undisturbed  by  it 

IRRATIONAL  EQUATIONS. 

32  !.  All  the  direct  methods,  employed  for  the  solution  of  equations  suppose 
that  the  unknown  quantities  in  them  aro  not  affected  with  any  radical  sign ; 
when,  therefore,  the  unknown  is  found  under  a  radical  sign,  it  will  be  neces- 
sary, before  applying  the  process  of  solution,  to  employ  some  preparatory 
method  of  rendering  the  equation  rational.  Such  a  method  is  at  once  sug- 
gested by  the  theory  of  elimination.  For,  if  we  equate  each  of  the  irrational 
terms  with  an  unknown  quantity,  and  remove  the  radical  from  each  of  these 
new  equations  by  involution,  we  shall  have  a  series  of  equations  (including  the 
original  one,  with  its  irrational  terms  replaced  by  the  new  symbols)  without 
radicals,  from  which  the  quantities,  temporarily  introduced,  may  Je  eliminated, 
and  thence  a  rational  equation  obtained,  involving  only  the  original  unknown 
quantities. 

The  following  examples  will  fully  illustrate  the  mode  of  proceeding : 

(1)  Let  the  equation  be 

.,_  ^/.r— 1+  y7+T=o. 

Put 


y=  y/x—  1,  z=  Vx+l\ 

and  we  then  have  the  three  following  rational  equations  from  which  wo  may 
eliminate  y  and  r,  viz., 

y*=x—l,  z»=a:+l,  x—y+z=Q. 

From  the  last  equation  we  gel  </=.'■-{-:,  and.  by  substituting  mis  value  in  the 
first,  y  becomes  eliminated,  and  we  have  these  two  equations  in  z  and  r,  viz., 

— X+l=sO 

4-0,.r  +  .r.  _,  +  !_(); 


EXPONENTIAL  EQUATIONS.  407 

and,  to  eliminate  z  from  these,  wo  apply  the  process  explained  in  the  preceding 
articles,  and  thus  get  the  final  equation 

x«  _  3.f5  _(_  8.x" + .t3 + 7.r2 — 7x + 2  =  0. 

(2)  Let  the  equation  be 

V4x+7+2Vi'-4  =  l- 
Putting 


7/=  V4x+7,  z=y/x— 4, 
wo  have  the  system  of  equations 

if=Ax+l,  z2=x— 4, 

EXPONENTIAL  EQUATIONS. 

325.  An  exponential  equation  is  an  equation  in  which  the  unknown  appears 
in  the  form  of  an  exponent  or  index ;  thus,  the  following  are  exponential  equa- 
tions : 

a*  =  b,  x*=a,  ah*=c,  x**  =  a,  Sec* 

To  resolve  the  equation 


10x=:2 


put  x=— ,  then 


10x/=2  .-.  10=2X'. 
The  value  of  x'  lies  evidently  between  3  and  4 ;  place  it,  therefore,  equal 
to  3  plus  an  unknown  fraction,  and  wo  shall  have 

i  i 

10=23+*",  or  10=23x2i" 

10        !  (by 

. ox"     .    1  —  1    — o 

"8 "     W    — A 

The  value  of  x"  lies  evidently  between  3  and  4,  •••  place 

'  and  proceed  as  before.     The  value  of  x  is  thus  obtained  in  a  continued  fraction. 

_1_1  _1      1 

which  may  bo  carried  to  any  extent  at  pleasure,  and  the  value  found  by  the 
method  explained  hereafter.     (See  Continued  Fractions.) 

When  the  equation  is  of  the  form  a*  =  b,  or  ah*=c,  the  value  of  x  is  readily 
obtained  by  logarithms,  as  we  have  already  seen  in  Art.  220.  But  if  the  equa- 
tion be  of  the  form  xx=a,  the  value  of  .t  may  bo  obtained  by  the  rule  of  double 
position,  as  in  the  following 

EXAMPLE. 

Given  £x=100,  to  find  an  approximate  value  of  x. 


*  Exponential  equations,  and  those  in  which  logarithms  of  unknown  quantities  enter, 
belong  to  a  claw  called  transcendental. 


408  ALGEBRA. 

The  value  of  x  is  evidently  between  3  and  4,  since  33=27  and  4*s=25G ; 
hence,  taking  the  logarithms  of  both  sides  of  the  equation,  we  have 

a- log.  x—  log.  100=2.* 


First,  let  £i  =   3-5;  then 

3-5  log.  3-5=    1-9042380 

true  no.  =   2-0000000 

error  =—-0957020 


Second,  let  x2=   3-G  ;  then 

3-G  log.  3-6=    2-0026890 

true  no.=    2-0000000 

error  =  +  -0020890 


Then,  as  the  difference  of  the  results  is  to  the  difference  of  the  assumed 
numbors,  so  is  the  least  error  to  a  correction  of  the  assumed  number  corre- 
sponding to  tho  least  error ;  that  is, 

•098451  :  -1  : :  -002G89  :  -00273; 
hence  x=3-6  — -00273=3-59727,  nearly. 

Again,  by  forming  the  value  of  xx  for  £=3-5972,  wo  find  the  error  to  be 
—  ■0000841,  and  for  x=3-5973,  the  error  is  + -0000149; 

hence,  as  -000099  :  -0001  : :  -0000149  :  -0000151 ; 
therefore,  x=3-5973  — -0000151=3-5972849,  the  value  nearly. 

EXAMPLES   FOR   PRACTICE. 

(1)  Find  x  from  the  equation  zx=5.  Ans.  2-129372. 

(2)  Solve  the  equation  x*  =  123450789.  Ans.  8-G400268. 

(3)  Find  x  from  the  equation  x't=2000.  Ans.  4-827822G. 

DEMONSTRATION  OF  THE  BINOMIAL  THEOREM. 

CASE  I. 

326.  If,  at  Prop.  V.,  Art.  245,  we  suppose  the  second  terms  au  a2,  a3,  &c,  of 

the  binomials  to  be  all  positive  instead  of  negative,  and  all  equal  to  a,  then  the 

products  two  and  two  will  all  become  a2 ;  those  three  and  three,  a3,  and  so 

on ;  and,  by  recurring  to  Art.  203,  we  perceive  that  the  number  of  combiua 

tions  or  products  two  and  two,  if  we  suppose  that  there  are  n  binomials,  will 

, ,     n{n  —  1)     ,             ,                     ,   ,              «(«  —  l)(n— 2) 
be  expressed  by  — — — ,  the  number  three  and  three  by — -— -,  and 

so  on.     Hence,  where  n  is  a  whole  number, 

(x+a)n=xn+naxu-l+-Y:^a-xa-°--\-,  &c,  -fa", 

or 

(a+»)n=an+nan-1z+Aan-2.t-+BuI:-:\r3+,  <5cc (1) 

Dy  reversing  the  order  of  the  terms,  and  disregarding  the  particular  form  of 
tho  coefficients  after  the  second  term. 

CASE    II. 

If  tho  exponent  bo  fractional,  wo  have 


(</  +  .r)»  =  V («+x)'"=  \Z«'"4-'"«'"  -lx+\W"   •.;-•+,  &c. 

*  In  equations  of  this  kind  the  following  method  may  be  adopted:  Let  x*=a  ;  then 
x  log.  x=  log.  a;  put  log.  x=y,  and  log.  a—/i ;  then  xy=b,  and  log.  x-f-  log.  y=  log  b; 
hence  y-\-  log.  y=  log.  b.     Now  y  may  be  found  l>y  double  position,  as  abovo,  and  then  r 

becomes  known.     When  a  is  less  than  unity,  put  sb=-  and  rt=- :  then  We  have  /<T=v 

h 

.-.  y  log.  })■=.  loir.  >/,  and  if  log.  b=C,  and  log.  V-  U  :  then  cy=Z.  and  log.  r-f-  Iolt.  _v=  ' 

or  log.  c-\-z=  log.  z.    Hence  -  mvv  be  found  by  the  preceding  method,  and  then  y  and  x 

become  known. 


DEMONSTRATION  OP  THE  BINOMIAL  THEOREM.  4CJ 

Applying  the  rule  at  Art.  113  for  extracting  the  root  of  a  polynomial,  the 

m 

first  term  of  the  root  will  be  a  °  ;  the  divisor  of  the  second  term  of  the  given 

(m\  m_m 

a*)      =na     a  ;  and  the  quotient  or  second  term  of  the  root 

will  be  -am~    ^m  *'x=-a*    x.    When  the  two  terms  of  the  root  thus  found 
n  n 

are  raised  to  the  rih  power,  and  subtracted  from  the  given  polynomial  accord- 
ing to  the  rule,  the  first  two  terms  of  the  latter  will  be  canceled,  and  the  next 

m— _        ...  ,  ra— 2 

highest  power  of  a  to  bo  divided  by  the  constant  divisor  na     u  will  bo  a 
multiplied  by  a?,  and  the  quotient,  which  is  the  third  term  of  the  root,  will 

contain  a  to  the  power  n— 2  —  (ra— J  = 2  with  a:2,  and  so  on,  so  that  the 

root  may  be  written  under  the  form 

m        jjj    2—1  —— 2  ™— 3 

a^-l—a."    x+A'an    x2+B'an    r'+.&c., 

the  same  form,  so  far  as  regards  the  exponents,  as  when  tho  exponent  is  a 
whole  number.  Tho  coefficients  will  be  examined  for  this  and  the  next  case 
together. 

CASE    III. 

When  the  exponent  is  negative,  either  entire  or  fractional,  as  a  consequence 
of  what  has  just  been  demonstrated,  we  have 

1 1 

But  if  the  division  bo  effected  according  to  the  ordinary  rules,  the  quotient 
will  be  indefinite,  and  of  the  form 

a-m—ma-m-lx-\-k"a-m~'1x2-\-,  &c.  ; 
then,  whatever  be  the  exponent  of  a  binomial,  its  development,  as  to  the  co- 
efficients of  the  first  two  terms  and  the  exponents  of  all,  is  of  the  same  form, 
viz.,  that  indicated  by  equation  (1). 

Now,  to  examine  the  coefficients  of  the  other  terms,  for  the  sake*  of  gen- 
erality, I  shall  consider  two  consecutive  terms  of  any  rank  whatever,  and  I 
shall  write 

(a + .r)m = am + ma,a~1x \-  Mam_n.T" + Nam-n_1a;n+1 + ,  &c. 

Let  us  change  throughout  x  into  x-\-y  ;  as  the  unknown  coefficients  con- 
tain neither  a  nor  x,  the  above  expression  becomes 

(a-\-x-iry)mz=am-\-mam-l(x-\-y) 

,    .+Mam-"(.r+i/)n+Nam-n-1(x-f-2/)n+1-f ,  &c. 
By  changing  a  into  a-\-y,  we  should  have  found 

(a+y+rp  =  («+?/)m+™(0+2/)m_I*- 
r-M(a+2/)m-n.in  +  N(a-f-2/)m-n-1xu+i4-,&c. 

In  the  two  proceding  equalities  the  first  members  are  equal,  therefore  the 
second  members  must  bo  equal  also  ;  and  this  is  the  case  whatever  values  x 
and  y  may  have.  Then,  if  they  bo  arranged  according  to  tho  powers  of  y, 
they  must  be  identical.  It  is  true,  they  contain  binomials,  but  we  know  the 
first  two  terms  of  each  of  these  binomials,  so  that  we  can  form  the  part  which, 
in  each  second  membei-,  contains  y  to  the  first  degree,  and  that  will  suffice  for 
our  purpose.  Designating  it  by  Yy  in  the  one  and  by  Y'y  in  the  other,  it 
is  easy  to  find 


410  ALGEBRA. 

Y  =mam~l r-Mnam-^-1  +  N(«--r-l)am-n-I.rB.... 

Y'=mam-1...  +M(»i  — «)am-n-'.fD+N(m  — n  — l)arr--°-V+ 

These  two  quantities  must  be  equal,  whatever  be  the  value  of  r ,•  the  co- 
efficients, therefore,  of  the  same  powers  of  x  must  be  equal.  Considering 
only  those  which  pertain  to  am-n~1x",  we  havo 

N(n  +  l)  =  M(»i— n) .-.  N  =  M("!~"\ 

We  see  by  this  according  to  what  law,  in  the  development  (1),  any  coeffi- 
cient whatever  is  formed  from  the  preceding.  It  is  the  same  that  we  have 
found  for  the  case  of  a  positive  exponent  (Art.  107,  IV.)  ;  and  as  we  have 
seen  that  the  first  two  terms  are  composed  in  the  same  manner,  whatever 
be  the  exponent  m,  it  will  bo  so  also  with  all  the  other  terms. 

An  abbreviate  notation,  sometimes  employed  to  express  the  coefficients  of 
the  binomial  formula,  is  the  initial  letter  B  of  the  word  binomial,  with  the  ex- 
ponent of  the  power  of  the  binomial  before  it,  and  the  order  of  the  coefficient 
above.     Thus,  the  coefficient  of  the  1°  term,  if  the  exponent  bo  n,  is  ex- 

0  1  2 

pressed  by   "B;'of  the   2°,    nB  ;   of  the   3°,   nB,    &c. ;    of  the    Z.-"1    term 

r»(»— l)...(n— k+1)         >?, 

— — — = by  nB,  or  otherwise  simply  nk. 


SERIES. 
RECURRING  SERIES. 

mill  a' 

327.  To  develop  the  expression         }     in  a  series,  place 

^=A+B.r+C^+)  ccc, 

ind  proceeding  by  the  method  of  undetermined  coefficients,  explained  at  Art 

209,  we  find 

a'  b  b  b 

A=-  B  =  — A,  C  =  —  B,  D  =  —  C,  ccc. 
a  a  a  a 

From  which  we  perceive  that  each  coefficient  is  obtained  by  multiplying  the 

b 
preceding  by  — -.     Hero  the  series  is  a  simple  geometrical  progression. 

Proceeding  in  a  similar  manner  with  the  fraction 
a'+bx 

we  obtain 

a'           b'  —  Ab  c        b  b 

A=-,  B= ,  C  =  — A-   B,  D= — B-  ■  C,  &c. 

a  a  a        a  a        a 

Hero  each  coefficient  from  tho  3°  is  tho  sum  of  the  twc  preceding,  multi- 

c  b 

plied  respectively  by  — -  and  — -,  or  each  term  is  the  sn  n  of  the  twi 

rr:  h  r 

ceding  multiplicil  by  — —  and  — — . 
1  J        a  a 

Again,  in  the  development  of 


RECUIUIENG  SERIES.  4U 

a'+b'x+c'x'2 


a-{-bx-[-cx2-\-dxi 

each  term  will  be  composed  of  the  three  preceding,  multiplied  respectively  by 

dx3       ex2        bx 

a  a  a 

finally,  it  becomes  now  evident  that  in  general  a  fraction  of  the  form 

a'  +  b'x+c'x2  .  .  .  +/t'xm-1 

a  -\-bx-\-cx~  .  .  .  -\-lc xm 

produces  a  series,  each  term  of  which  from  the  (m-\-l)a  is  composed  of  the 

Jc            Ji                       c           b 
m  preceding,  multiplied  respectively  by  — -xm,  — -xm~l,  .  .  .  — -x2, x. 

Sei'ies  of  this  form  are  called  recurrent,  and  the  assemblage  of  quantities  by 
which  it  is  necessary  to  multiply  several  consecutive  terms  to  obtain  the  fol- 
lowing term,  is  called  the  scale  of  relation  of  the  terms. 

328.  Problem. — A  recurring  series  being  given,  to  return  to  the  generating 
fraction. 

In  this  enunciation  it  is  supposed  that  the  recurring  series  is  arranged  with 
respect  to  an  indeterminate  x.     Let 

S=A+Bx+Cx2-f-.  .  . 
be  such  a  series,  having  for  a  scale  of  relation  [px3,  qx2,  rx~\.     Since  this  scale 
contains  three  terms,  the  generating  fraction  is  of  the  form 

a'+b'x+c'x2 

a  -f-  bx  -\-  ex2  -f-  dx3' 

If  this  fraction  had  been  given,  wo  have  seen  that  the  scale  of  relation  would 

r    d  c         b  ~\ 

bo x3,  — -x2,  — —x  .     But  the  generating  fraction  can  be  written  thus, 

a'     b'       c' 

— +-.T+-Z2 

a  '  a     'a 


bed' 
14—X+-X-+-X3 
'  a       a      'a 


and  then  we  perceive  that  the  three  terms  in  x  of  the  denominator  can  be  at 
once  obtained  by  taking  those  of  the  scale  of  relation  with  contrary  signs. 
Thus,  we  can  put  the  generating  fraction  under  the  form 

a  +  j3x+yx°~ 
1  —  rx  —  qx'2  — px* 
and  wo  shall  only  have  to  determine  a,  /?,  y.     To  do  this,  place 
a  _i_/j.r  J_  yx2 
1 — rx — qx2 — px3 
and  since,  after  clearing  it  of  fractions,  the  equation  ought  to  be  identical  in 
form,  we  derive  from  it,  having  regard  only  to  the  first  three  terms, 

a+/?x+7.r2=A+B    x+C    x8 
—  Ar    —Br 

-\q         ' 

Consequently,  we  shall  have  for  the  generating  fraction 

A  +  (B  —  Ar)x+  {c—Br—\q)xi 


S=- 


1  — rx  —  qx3 — px 


■  •■ 


412  ALGEBRA. 

For  example,  let  S  =  l — 2r — .v; —  .">,-{- 4. r4 — ...  he  a  recurring  series, 
whose  scale  of  relation  is  [-{-x3,  +4a?,  — 2x],  Taking  the  above  formula,  we 
shall  have 

A=l,  B  =  — 2,  c=  —  l,p  =  l,  ?=4,  r=— 2, 
%nd  we  shall  find 

1  +  2j— 4x-— a-3' 

329.  Problem. — A  scries  being  given,  to  determine  whether  it  be  recurring, 

and,  in  this  case,  to  return  to  the  generating  fraction. 

Let  the  given  series  be 

S=A+B:r-fCx2-f-L\rH 

a' 
Let  us  determine  first  whether  it  be  equal  to  a  fraction  of  the  form  — r-r-. 

1  a-\-bx 

a' 
and  place  S=        ,    .     From  this  equation  we  derive 

1      a-\-bx      a       b 

S=~aT~=a'Jt~a'X; 
the  quotient,  therefore,  of  (1),  divided  by  the  series,  ought  to  be  exact,  and  of 
the  form  p  +  ax'     Then  the  generating  fraction  will  be  expressed  thus  : 

P  +  <lx 
If  the  division  does  not  stop  at  the  second  term  this  series  will  not  be  recur- 
ring, or  else  it  will  arise  from  a  more  complicated  fraction. 

a'4-b'x 

Place  S= — r-7 — ; ;,  we  shall  have 

a-\-bx-\-cx' 

1      a_|_6.r-f  c.r2  a"x2 

S=~aT+b;x~  =P+qX^"a^+Vx ; 

that  is  to  say,  dividing  (1)  by  the  series  S,  if  wo  stop  the  division  after  wo 
have  obtained  as  a  quotient  terms  of  the  form  p-\-qx,  the  series  Si.r:,  which  is 
the  remainder  that  we  then  have,  and  which  is  always  divisible  by  a-,  will  bo 

Si  a" 


such  that,  after  we  have  removed  this  factor,  wo  must  havo 

^h  +  qtx; 


S~~a'+b'x' 
Hence  wo  derive 

S      a'+b'x 


S~     a' 

that  is  to  say,  the  new  division  ought  to  terminate  at  tho  socond  term  in  the 
quotient ;  and  then,  to  find  tho  generating  fraction,  wo  shall  have  the  two 
equations 

1  S,       S 


g=p  +  <7.r+-g.r3,  g-=p1  +  71r, 


whence 


.       ,  S,   '  S-jp.+jiX 

Consequently,  tho  generating  fraction  will  be 


{]>  +  <! '■)(!''  + 'h'l+x* 


RECURRING  SERIES.  413 

Suppose  that  the  quotient  of  S  by  Si  is  not  exactly  pt  -L.  71X ;  if  the  series 
e  recurring,  it  will  bo  of  an  order  superior  to  the  second.     Let  us  examine  if 

a'+h'x-lr-c'x* 

we  can  have  S  =  — —, — : — — — r^- 
a-\-  bx-{-cxi-\-dxi 

We  derive  from  this  equation 

1.  a"+b"x 

S=P+?X+ a'+b'x+c'x^ ; 

that  is  to  say,  after  having  obtained  the  first  two  terms  of  the  quotient  of  1, 
divided  by  the  series  Si,  wo  shall  find  for  a  remainder  a  series,  all  of  whose 
terms  will  contain  z2 ;  and  if  wo  designate  this  remainder  by  Six2,  we  shall 
have 

Si         a"+b"x 


This  equality  gives 


ti—a'+b'x+c'x- 


-r=pi  +  ?1x+ 


,-.• 


hence,  designating  by  S3.t2  the  series  which  we  find  for  a  remainder  after 
having  earned  the  division  of  the  series  S  by  the  series  Si  to  the  terms  of  the 
quotient pi-\-qiX,  we  should  have 


Sx— a"+b"x 
From  this  last  equality  we  derive 

■^-=p.-\-q.1x. 

Here  the  operations  stop  ;  for,  returning  to  the  generating  fraction,  we  shall 
have  the  equations 

1  S,       S  Sa      Si 

S=P:H'r+g-*2>  ■Q—Pi  +  qiX+~x'i,  ■^-=p.2+q2x; 

and  from  these  equations  we  derive 

IS,  1  S,  1 


S: 


.        .  Si    '   S  S2    '  Si     pn+qjx 


We  have,  then,  only  a  fow  substitutions  to  make  in  order  to  obtain  a  frac- 
tion equal  to  S. 

Without  proceeding  further,  the  reader  will  doubtless  perceive  that  the 
successive  operations  for  finding  the  quotients  jp+<p,  pi-\-qiX,  &c,  and  for 
returning  to  the  generating  fraction,  bear  a  striking  analogy  to  those  which  are 
necessary  in  reducing  an  ordinary  fraction  to  a  continued  fraction,  and  in  re- 
turning to  the  ordinary  fraction.  This  observation  will  take  the  place  of  a 
general  rulo.  If  we  arrive  at  a  division  which  gives  an  exact  quotient  ot  the 
form  p+5-r,  we  know  that  the  series  is  recurring.    (See  Contin.  Fractions.) 

EXAMPLE. 

Suppose  we  wish  to  determine  Whether  the  series  of  numbers  1,  2,  3,  &c, 
be  recurring.  It  is  not  this  numerical  series  which  we  must  consider,  but  the 
equation 

S  =  l+2a:4-3x24-4r?+  . . . 

We  perceive  that  the  operations  will  be  performed  as  follows  : 


414  ALGEBRA. 

Division  of  1  by  S 
1  n_|_2:r-f.3.r2-r-4x3-|- 


l  +  2j  +  3.r8+4r'4- |1  — 2x 

— 2x — 3x- — 4iJ — bx* —  .... 

— 2x— 4x-  —  Gx3  — 8x*  — 

x2+2x3+3x4+ =  Si.r3 

Division  o/S  by  Si. 


l-4-2x  +  3.r=4-4r»+ 
1  _|_  o.r  _j_  3X2+4X3 + 


1  +  2  •-|-3x'4-4.r54- 
1~ 


0 

l  s,.s 


Hence,  the  series  S  is  recurring,  and  we  have  ^  =  1— 2x-|--^£21  oi  =  l- 

IS                                                     1 
Wo  derive  from  this  S= 5 ~-  =  l ;  consequently,  S=- — : — : 

l_2.r+-rx3 


(i-xy 

Remark. — In  finding  a  rule  to  determine  whether  a  series  is  recurring,  we 
have  considered  the  series  as  derived  from  a  fraction  whose  numerator  is  of  a 
degree  inferior  to  the  denominator.  But  even  if  this  last  condition  does  not 
have  place,  it  is  easy  to  perceive  that  the  same  explications,  and,  consequently, 
the  same  rule,  will  always  subsist. 

329.  Problem. —  To  find  the  general  term  of  a  recurring  series. 
Suppose  that  the  series  has  for  a  generating  fraction 

a/+6'a+ j-frz"1-1 

=  a-\-bx  + \-kx'u 

We  can  write  this  fraction  thus  : 

F=(a'jf  b'x.. .  +h'xm-1){a+bx+ \-kx™)--. 

It  is  evident,  then,  that  by  developing  the  power  — 1,  ob^-r.ning  the  product 
of  the  two  factors  in  this  equation,  and  taking  in  this  product  the  part  which 
contains  x  to  any  power  whatsoever,  we  shall  have  the  general  term  of  the  re- 
curring series.  But  the  problem  is  resolved  ordinarily  by  another  process, 
which  I  proceed  to  exhibit. 

Wo  divide  first  all  the  terms  of  the  fraction  F  by  k,  and  write  it  under  the 
form 

U_     a'xm-l+!J'xm-*+  . . . 
y— a.m_|_flrn-i_|_^l.„   a_^_  _  _  _• 

The  fraction  is  supposed  in  all  cases  to  be  reduced  to  its  most  simple  form, 
so  that  U  has  no  common  factor  with  V. 

Wo  decompose,  then,  the  denominator  into  binomial  factors,  such  as  X-\-a, 
whether  it  be  by  equating  this  denominator  to  zero,  or  by  some  other  method, 
and  then  the  fraction  is  regarded  as  resulting  from  tho  addition  of  many  others, 
which  have  for  denominators  these  different  factors.  We  determine  all  these 
partial  fractions,  anil  then  form  the  general  term  of  tho  development  of  each  : 
then,  taking  the  sum  of  these  general  terms,  we  shall  have  the  general  term 
of  the  recurring  series. 

[n  this  decomposition  into  partial  fractions  it   is  necessary  carefully  to  dis 


SUMMATION  OF  SERIES.  415 

tinguish  in  V  the  simple  faoiors  from  those  which  are  raised  to  powers  For 
each  simple  factor  x-\-a  we  shall  takea  fraction  of  the  form 

M 
x-\-a 
For  each  factor,  such  as  (r-f  h)n,  wo  might  take  one  of  the  form 

Ax"-1  +  B.r'-24-... 
(x+b)n  ' 

but  it  is  more  convenient  to  have  only  fractions  with  monomial  numerators ; 
instead,  therefore,  of  a  fraction  like  the  preceding,  we  take  n,  like  the  fol- 
lowing : 

N  N,  N2  Nn_x 

(x-\-b)n~i~  {x+b)n~l  +  {x+b)«-2        *~x_^b> 

M,  N,  Ni...  representing  quantities  independent  of  a:. 

Consequently,  if  we  suppose  that  V=(.T-f-a)(j-'+&)n-  ■ .,  wo  can  place 
U        M  N  N.  Nn_, 

V~ x+a+(r-r-i)"+(z+i)"-1 ' ' '  +x+b^       ' 
and  the  question  will  be  reduced,  for  the  present,  to  the  determination  of  the 
numerators  M,  N,  Nlt  &c.     But  these  have  been  determined  in  Art.  209,  (3) 

The  preceding  decomposition  being  effected,  the  determination  of  the  gen 
oral  term  of  the  recurring  series  does  not  offer  any  difficulty. 

Each  partial  fraction  can  be  put  under  the  form  P(p-\-x)~^,  designating  by 
H  an  entire  positive  number,  which  can  be  equal  to  1.  If  we  develop  thi? 
power,  we  readily  find  that  the  term  affected  with  xa  is 

1 .  2  .  3  ...  n  rp 

It  is  the  sum  of  like  expressions,  all  containing  xn,  and  resulting  from  the 
different  partial  fractions  which  compose  the  general  term  required. 

When  the  denominator  of  the  generating  fraction  contains  imaginary  fac- 
tors, these  factors  introduce  imaginary  quantities  into  the  general  term.  If 
we  suppose,  however,  that  the  coefficients  of  the  numerator  and  denominator 
of  the  proposed  fraction  are  all  real  (and  they  are  always  taken  so),  it  is  evi- 
dent, a  jniori,  that,  as  we  find  the  development  of  this  fraction  by  division,  the 
general  term  can  not  embrace  any  imaginary  factors ;  consequently,  wo  are 
sure  that  all  the  imaginary  quantities  which  arise  from  the  factors  of  the  de 
nominator  will  disappear. 

SUMMATION  OF  SERIES. 

The  summation  of  series  is  the  finding  of  a  finite  expression  equal  to  the 
proposed  series,  even  when  the  series  is  infinite,  and  in  many  cases  this  finite 
expression  is  found  by  subtraction. 

EXAMPLES. 

Ill  . 

(1)  Required  the  sum  of  the  series  ttj+fTq  +  o"!-!"  •  •  •  •  t0  infinity. 


ad  infinitum. 


„     1     1     1     1     1     1 

LetS  =  l  +  2+3+4  +  5+G+ 

111111  ,.  , 

.-.  S  —  l=-+-+-+g+-+-+ ad  infinitum. 


416  ALGEBRA. 

Hence,  by  subtracting  the  latter  from  the  former,  wo  have  the  required  sum 
11111 
1^+2^3  +  31+4^5  +  5.0  + 

1         1         1 

(2)  Required  the  sum  of  the  series  7-^+-rr+TT+ to  n  terms. 

l.o        —.4        o.O  a 

1111  1 

LetS=I  +  2+5+^+ -         (a) 

1111111  1 

-  S-1-2+^H  +  ^+2=3+4  +  5  +  G+ Z+2 <*> 

Subtracting  (b)  from  (a),  we  have 

11  12         2         2         2  2 

1  +  2_  ?i+l_  7i-f2=r3+2T4+3T5+4l;+ n(w+2) 

J_       1        J_      J_  _1_      _1$         1/1  1     \  > 

•''  1.3+2T4+3^+4.G+"7i(«+2)~2^1  +  2— Vi+1  +  71+2/  $ 

~2t  1~~7i-r-l  +  2_  n  +  2> 

n  n 

=  2ft+2+4n+8' 

Wlien  n  is  infinitely  great,  then  wo  have 

1111  ,  .  „  .  1/       1\      1       1      3 

t-^A-^-t-\-7tt+t^+  ...  ad  infinitum  =-(  1  +  -)  — —  =7. 

1.3~2.4~3.5~4.6  '  2\     '2/       co      oc      4 

1111 

(3)  Sum  the  series  7^—777+^— TT+ ad  infinitum. 

Ans.  1 
4 

(4)  Sum  the  series  7^+0^X5+3^X6+ ad  iQfinitura' 

Ans.  jg. 

e  f?  17 

(5)  Sum  the  series  7^3+2^4+3X5+ t0  n  termS" 

3        2  1 

Ans.  7:  —  — TT  +  — TTi- 
2      71+1  '  7»  +  2 

(6)  Sum  the  series  a-f- 2ar4-3ar2+4ar3-f-  .  ...  to  71  terms. 

f    1 — r°        nra   ) 

Ans.  a  <  yz rz — - ?  . 

I  (1— r)s     1 — r  > 

(7)  Sum  the  series  l  +  3r+5x2-r-7.r34-9x4  ....  ad  infinitum. 

An9-  $=& 

DIFFERENCE  SERIES. 
330.  Let  there  be  the  ai-ithmotical  progression 

a,  a  +  c5,  a  +  2t,  a  +  36 

If  wo  begin  with  a  now  term,  b,  and  add  to  it  successively  each  term  of  the 
above,  wo  obtain 

b,  6-fa,  b+2a+t,  b+3a+36,  &  +  4a+GJ .  . ., 
which  is  called  a  difference  series  of  the  2°  order,  and  so  on,  as  in  tho  follow 
ing  scheme  : 


DIFFERENCE  SERIES.  417 

Merles0'    1C  teTm-    2°  *'"'""■  3°  '*""  ""'  'er:D' 

"l        a,       a+d,       a-\-2d,       .  .  .  a+(n—l)6. 

H.        b,        b+a,        b+Za+6  .  .  .  b-\-{n— 1)«+(W~^    "- rf. 

.„..                 .,        iw   ,  (a— S)(it— 1)_  ,  (»-3)(«-2)(k— 1), 
III.        c,        c+4        c+2b+a  .  .  .  c+(»— 1)H — «+  x   2  ~3 * 

tec.       &c. 

EXAMPLE. 

I.  order,  2,  5,   8,   11,  14  .  . 

II.  order,  4,  6,  11,  19,  30  .  . 

III.  order,  5,  9,  15,  2G,  45  .  . 

331.  From  the  manner  in  which  these  difference  series  are  formed,  it  is 
evident  that  if  we  subtract  from  one  another  the  successive  terms  of  any  or- 
der,  we  obtain  the  terms  of  the  preceding,  and  continuing  i.i  this  way  till  wo 
subtract  the  successive  terms  of  the  first  from  one  another,  we  obtain  between 
them  the  constant  difference  6. 

332.  If  the  order  of  a  series  be  unknown,  its  order  may  be  found  from  what 
has  been  said  above.     Thus  the  series 

5,  9,  15,  26,  45  , 
taking  the  difference  of  the  consecutive  terms. 

4,   6,   11,  19 

2,  5,     8 

3,  o,     3, 

after  three  subtractions  of  consecutive  terms  presents  a  constant  differenco, 
and  is,  therefore,  a  series  of  the  3°  order. 

333.  To  separate  the  roots  of  an  equation  by  means  of  difference  series. 
The  xlh  term  of  a  series  of  the  order  m  would  be  expressed  by 

*+(*—!)/+ x  .  2         g+  ' '  •  •  +  1 .  2  . . .  m  ' 

which,  arranged  according  to  the  powers  of  x,  would  be  of  the  form 

M;rm4-A.rm-1  +  B.rm-3  ....  +Gx+K; 
that  is,  of  the  form  of  the  first  member  of  an  equation  of  the  mtU  degree,  X=C. 
If,  now,  we  give  to  x  the  values  . .  .  — 4,  — 3,  — 2,  — 1,  — 0,  1,  2,  3,  4, ... . 
representing  the  values  which  the  polynomial  X  assumes  by 

X_4,  X_3,  X_2,  X_i,  X0,  Xi,  X2,  &c (1) 

these  quantities  will  form  a  difference  series,  since  x  denotes  the  order  of  the 
term  in  a  series  of  which  X  is  the  general  term.  There  is  no  objection  to  x  being 
negative,  as  a  series  may  be  continued  below  as  well  as  above  the  first  term, 
observing  the  same  law  in  a  contrary  sense. 

Taking  a  sufficient  number  of  terms  of  the  series  (1)  to  obtain,  by  subtrac- 
tion of  its  successive  terms,  the  series  of  next  lower  order,  and  from  this,  in 
the  same  manner,  that  of  the  next  lower  order  still,  till  we  arrive  at  constant 
differences,  the  terms  of  the  series  (1)  may  be  extended  indefinitely  to  the 
right  and  left  by  forming  them  according  to  (Art.  330),  without  the  trouble  ot 
substituting  numerical  values  for  X,  and  calculating  the  corresponding  values 
of  X.  Those  values  of  X  which  have  contrary  signs  will  (Art.  252,  Cor.  1) 
have  one  or  an  odd  number  of  roots  between  them. 
Take,  for  example,  the  equation 

9.r4_3.c3_i30x2_l7.r+260=0. 
D  D 


418  ALGEBRA. 

Giving  x  the  values  — 2,  — 1,  0,  1,  2,  we  have  the  following  results  inc'os*u. 
in  the  parentheses  : 

j£._4    -X._3      X_j      A — i        Xo         A.)         A:        A3       A_t 
+  744_  49(_  58  -J-159  +  260  +119  —174)— 313+224, 

forming  a  series  of  the  fourth  order.     The    eriea  of  the  third  order  is 
—793  —     9(  +  217  +101  —141  —293)  — 139  +  537  ; 
A  the  second,  +784  +  226(  — 116  —242  — 152)+154+67l 

of  the  first,  —558  — 342(— 126  +   90) +306+ 522  ; 

equal  differences,  +216  +216(+216)+216+216. 

By  substituting  other  values,  as  —3,  — 1,  —5,  — G,  and  +3,  +4.  +  5,  +6, 
&c,  we  may  extend  the  top  series  to  any  length. 

To  save  the  time  and  trouble  of  substituting  consecutive  numbers  and  calcu- 
lating the  result,  the  method  of  difference  series  is  employed,  thus  : 

Substitute  a  number  of  consecutive  values  one  more  than  the  degre 
equation;  the  smallest  numbers,  being  more  easily  substituted,  are  preferred. 
In  the  present  example,  substituting  — 2,  — 1,  0,  1,  2,  we  obtain  thai  pi 
of  the  first  scries  which  is  of  the  3°  order,  included  in  brackets  ;  from 
by  subtracting  its  consecutive  terms,  the  corresponding  portions  of  the  s 
of  the  2°  order,  and  so  on  ;  and,  finally,  the  diiference,  216.     Using  this  dif- 
ference, we  may  extend  the  top  series  at  pleasure,  according  to  the  methud 
in  Art.  330. 

The  roots  of  the  equation  lie  between  those  numbers  the  substitutions  of 
which  produce  unlike  signs  in  the  result ;  thus,  in  the  above  there  is  one  root 
between  — 3  and  — 4,  one  between  — 1  and  — 2,  one  between  1  and  2,  and 
one  between  3  and  4. 

334.  There  exists  between  the  coefficients  of  two  consecutive  powers  of 
r+rt  relations  from  which  many  useful  consequences  may  be  deduced. 

Suppose  the  mlil  power  of  x+a  to  be 

xm+Aaxm-1+Ba2xm-2+Ca3x,n-3+,  Sec. 
Multiplying  the  polynomial  by  x+a,  there  results 

xm+i  +  A«xm  +  Ba2xm-1  +  Ca3xm-2+  .  .  . 
+    axm+Aa  x^  +  Ba  xm-2+  .  .  . 

From  which  we  conclude  that,  to  ohtabi  the  coefficient  of  any  term  of  the 
(m  +  iy*  power  q/"x  +  a,  it  is  only  necessary  to  add  to  the  coefficient  of  the  term 
of  the  same  rank  in  the  m'h  power  that  of  the  preceding  term. 

335.  According  to  this  rule,  we  can  form  the  coefficients  of  the  sucw»«s!vo 
powers  of  x+a,  as  may  be  seen  in  the  following  table  : 

1,  1,   1,  1,   1,1,     I,     1,     1    .  .. 

1,  2,  3,  4,    5,     6,     7,     8   .  .  . 

1,  3,  6,  10,  15,  21,  23  .  .  . 

1,  4,  10,  20,  35,  E6  .  .  . 

1,    5,    15,  35,  70  .  .  . 

1,     6,    21,  56  .  .  . 

1,     7,    28  .  .  . 

1,     8   .  .  . 

1    .  .  . 

The  first  vertical  column  of  this  tablo  is  formed  of  the  single  number  1.  The 
•econd  column  is  formed  of  the  number  1  written  twice.      We  fbfl)*  the  third 


THE  DIFFERENTIAL  METHOD  OF  SUMMING  SI  IUES.  119 

column  by  placing  at  the  side  of  each  term  in  the  second  column  the  number 
obtained  by  adding  it  to  the  term  above  it;  we  find  thus,  for  the  first  term  of 
the  third  column  1  +  0  or  1>  tne  second  term  is  1-f-l  or  2,  and  the  third 
0  +  1  or  1.  The  fourth  column  is  deduced  from  the  third  in  the  same  manner 
that  that  is  from  the  second,  and  so  on.  The  two  terms  of  the  second  column 
may  be  considered  as  the  coefficients  of  the  first  power  of  :r+«.  It  results 
from  the  above  rule  that  the  terms  of  the  third  column  are  the  coefficients  of 
the  development  of  (ar-j-a)3,  those  of  the  fourth  column  of  (■•■'+ '')\  &c. 

This  tabic,  which  may  be  indefinitely  extended,  is  called  the  Arithmetical 
Triangle  of  Pascal. 

336.  It  is  easy  to  see  from  the  composition  of  the  arithmetical  triangle  that 
the  pth  term  of  any  horizontal  line  is  the  sum  of  the  p  first  terms  of  the  pre- 
ceding horizontal  line.  Because  if  we  consider,  for  example,  the  term  56, 
which  is  the  sixth  of  the  fourth  line,  this  term  is  formed  by  adding  the  two 
numbers  21  and  35,  which  are  placed  at  its  left  in  the  third  and  fourth  lines; 
but  the  second  of  these  two  numbers,  35,  is  the  sum  of  15  and  20  ;  the  last 
number,  20,  is  the  sum  of  10  and  10,  and  the  last  number,  10,  the  sum  of  6 
and  4 ;  finally,  4  is  the  sum  of  the  two  numbers  3  and  1 ;  we  have,  therefore, 
56  =  21  +  15  +  10  +  6  +  3  +  1. 

THE  DIFFERENTIAL  METHOD  OF  SUMMING  SERIES 

337.  Let  a,  6,  c,  d,  e, . .  .  .  be  a  series  of  terms,  in  which  each  term  is  less 
than  the  succeeding  one  ;  and,  taking  the  successive  differences,  we  have 

c  d  e,  Sec. 

c  —  b  d — c  e — d,  &c. 

■  2b+a  d  —  2c+b  e—2d+c,  Sec. 

d— 3c+36— a  e—3d+3c—b,  &c. 

e  —  4o?+6c  —  46+a,  Sec. 

Putting  dii  d»,  d3,  d+, for  the  first  terms  of  the  first,  second,  third 

fourth, ....  differences,  we  have 

b —  a  =d1  .-.  b=a-{-  dx 

c — 2b-\-a  =d2  .'.  c=a+2c?1+  d.t 

d—3c-\-3b—a         =d3  .-.  tZ=a  +  3d1+3rf2+  d3 
e — id-\-6c — 4b-\-a=d4  •'•  c  r=«  +  4d,  +  6c£2+4<f3+(::0 
&c.  &c. 

Hence  the  (n+1)"'  term  of  the  proposed  series  is  evidently 

.     A   .     fr-1)^    ,  »(n-l)(n-2), 
a+ndl+n~Y^-di+ 1.2.3 *3+ 

and,  therefore,  the  nth  term  is  (by  writing  n  —  1  for  n) 

n±t„      iw  ..  (n-l)(n-2)         (B-i)(w-2)(n-3) 

338.   To  find  (S)  the  smn  of n  terms  of  a  series. 
Let     a,  b,  c,  d,  e,  Sec. 

and  0,  a,  a-\-b,         a-\-b-\-c,  <z  +  i  +  c+d,  &c., 

be  two  series,  of  which  the  («.  +  l)th  term  of  the  latter  is  obviously  the  sum  of 
n  terms  of  the  former:  but  the  first  terms  of  the  first,  second,  third,  fourth 
....    differences  in  the  latter,  are 


a 

b 

(dx) 

b—a 

(A) 

c 

(d3) 

(«*0 

420  ALGEBRA. 

a,  b — a=di,*  c — 2&  +  fl  =  </;,  d — or.-\-%b — a,  =dz,  &c  ; 
honce  the  (tt  +  l)th  term  of  the  latter  series,  or  the  sum  of  n  terms   if  tb 
former,  is,  by  (1)  in  the  last  article, 

^      ~     1.2  l~        1.2.3          *~            1.2.3.4             3~        ' 
or 

n{n  —  1)  n(n  — 1)(»  — 2)  ,      n(n—l)(n— 2)(n  — 3)  , 

T     1.2  ^         1.2.3          -^            1.2.3.4            "i-r-'-l" 

EXAMPLES. 

U)  To  what  is  1.2+2.3+3.4  +  4.5+  . . .  n(n+l)  equal? 
2,  6,  12,  20,  30,  is  the  given  series  ; 

4,  6,    8,   10,  differences  of  the  consecutive  terms; 
2,    2,    2,  differences  of  these  again,  d2 ; 
0,    0. 
Hence,  a=2,  di=4,  d2=2,  and  d3,  d4,  <kc.  =0;  therefore 

<5      --  ■  W^n~1).i    .  *»(n— 1)(n— 2) 
S=na+ - d,+ — d3; 

=2n+2»(n—  l)+«n(«  —  l)(n— 2) 
=  I«(n+l)(n  +  2). 
Proceed  always  in  this  way  till  the  differences  become  the  same.f 

(2)  Find  the  sum  of  n  terms  of  the  series  1,  23,  3',  43,  53,  &c. 

(3)  Find  the  sum  of  n  terms  of  the  series  1,  4,  10,  20,  35,  &c. 

(4)  To  what  is  1.2.3+2.3.4  +  3.4.5+ n(»+l)(»+2)  equal?      ■ 

(5)  Sum  n  terms  of  the  series  1,  3,  5,  7,  9,  11,  &c.  . . . 

(6)  Find  the  sum  of  15  terms  of  the  series  1,  4,  8,  13,  19,  &c 

(7)  Sum  8  terms  of  the  series  1,  24,  3\  44,  54,  64,  ecc. 


(2) 
(3) 


ANSWERS. 

n"(n-\-lf 
4        ' 
n(n+l)(»+2)(n  +  3) 


1.2.3.4 
(4)  >(n  +  l)(n+2)(n+3). 


(5)  n*. 

(6)  Jn(n«+6n  — 1)=785. 

n5     n*     n3      n 

(7)  T+-+^-rn=8772. 


5   '    2  t    3      30 

POWERS  OF  THE  TERMS  OF  PROGRESSIONS 
339.  If  all  the  terms  of  a  geometrical  progression 

-ffa :  aq :  aqz :  ag8 aqn~x 

are  raised  to  the  same  power  m,  the  result  is  the  series 
am,  amqm,  amq"m,  a™^"1 aaf^-l\ 

which  is  a  geometrical  progression,  of  which  the  first  term  is  am,  the  ratio  qm, 
and  the  number  of  terms  n. 

>40.  If  the  terms  of  11  p  ion  by  differences,  whose  first  term  is  a  and 

■'minion  difference  <*,  bo  each  raised  to  the  »r   power,  we  ha^e 

*  Tliis  is  the  <h  of  the  former  series,  but  the  dg  of  the  latter. 

t  The  terms  of  tlio  formula  ('J),  containing  those  orders  of  differences  which  becomo  zero, 

like  ''.!,  </i.  i\i-.,  iu  1  .-mi]il.'  l,  will  all  vanish,  ami  tin ■  expression  for  S  will  be  composed 
imly  of  the  prec       Dg  I    rmfl. 


POWERS  OF  THE  TERMS  OF  PROGRESSIONS.     -  421 

am±E:am. 

m(m — 1) 
(a+   d)m=am+mam-1  6+  0— Qm~2  (P+,  &c. 

JL  •  A 

m(m — 1) 
(a— 2i5)m=am+wiam-12<5+— j-^—!-am--U2'-\- ,  &c. 

ra(m — 1) 
(a— 3J)m=ara+mam-13(J+-Y-2-iara-29(52+,&c. 

&c.  &c. 

Taking  the  differences  of  the  consecutive  terms, 

m(m  —  1) 
(a+  6)'"—  am  =mam-l6-\-  V"-^8  +  ,  &c. 

(a+2(5)m  — (a+  (3),n=mam-M-{-     ^~    am-33<P+,  &c. 

(a+3<5)m— (a  +  2(5)m=mam-M4-?^Y^^ara-25(32+,&c. 

These  differences  being  not  the  same,  the  same  powers  of  the  terms  ol   »n 

arithmetical  progression  do  not  form  an  arithmetical  progression. 

341.  To  find  the  sum  of  the  mth  powers  of  an  arithmetical  progression.    Let 

—a  .6  .c.d.  ..Te.l 

be  any  arithmetical  progression,  of  which  the  common  difference  is  <5.     Then 

b=q+3,  c=b+6, l=Jc+6. 

Raising  these  equalities  to  the  power  m-{-l, 

,    .  ,  (m-\-l)m 

b'n+1=zam+l  +  (wi  +  l)am<J+1-y L-^L-am-162-^,  &c. 

X  •  A 

+1==&ra+1+(m+l)imd+^i-^-V-1(52+,  &c 


\, 


C 


/m+1==^>+1  +  (m  +  l)7cmJ+^Y1J—  &™-M*+,  &c. 

Adding  all  these  equalities,  suppressing  the  common  terms  in  the  two  equa 
sums,  viz.,  bm+1,  cm+1,  &c,  and  transposing  am+1,  we  have 

Jrn+l  _am+l  _  (m_i.  l),5(am_L-  ft-  ..  .  .  _j-  &»), 

-f^—^—  6*(am-1  +  bm~l . . . . +7cm-1), 

+  ,&c. 
1  o  abridge,  let 

a  +6  +c+d \-Tc  +1  rsSn 

a2 +6* +Ar3+p.=:Ssi 

G«>_L.6m^_ +  £m+ Zm=Sm. 

Then  the  last  expression  becomes 

The  value  of  Sm  deduced  from  this  is 

l<"+i—.am+l     m  m(m—l) 

Sm==^+^?Iq:i^-^(Sm_I-^->)--^-'<5HSra_2-Z'«-2)-,  &c.    (1) 

The  law  of  the  unwritten  terms  is  sufficiently  apparent,  and  the  series  must 
evidently  end  with  the  term  preceding  that  which  contains  the  fac':or  ri — m 
or  0. 


422  ALGEBRA. 

By  formula  (1)  the  sum  Sm  can  be  found,  when  the  sums  of  the  inl  >r.ot 
powers  are  known;  for  this  purpose,  make  ?n  =  0,  the  formula  gives  S0 ; 
making  m  =  l,  it  gives  S^  and  so  on  to  the  sum  of  the  powers  required. 

If  the  progression  -4-a.a  +  <5.a  +  .2'5 is  replaced  by -^-1 .2.3 N  (or 

the  series  of  natural  numbers  from  1  to  N),  i.  e.,  a  =  l,  6=1,  i  =  N,  then  for- 
mula (1)  becomes 

Nm+l  —  1      m               „            m(m  —  l) 
S'"  =  N°+-^+r-2(Sm-1-Nm_I) -273     (S^-N™-2)-,  &&       (2) 

If  m=0,  (2)  becomes 

N°+'  — 1            N  — 1 
S0=N°+— —  =  1+-^ =N (3) 

N(N+1) 
Sl~        2  (4) 

S9=N*+^=i-(S1-N)-i(S0-No), 


Ifmrrl, 
Ifm=2, 


3  v    '         '      3V 


:N* 


N3— 1      /N2+N        \      1 
f-—  —  (-J—  Nj-3(N-1), 


XT      N3     1     N*     N     xr     N     1 
=  N2+3-3-T-2+N-3+3' 
_N3     N«     N_2N»+3NS+N 
— ~3^~Y^  ~6~~  6  ' 


N(N+1)(2N  +  1) 

&2=_  ~6 (5j 

formula  (3)  expresses  the  sum  of  l°+2°+3° to  N  terms,  or  of  14-1 

-f-1...  to  N. 

EXAMPLES. 

(1)  If  7n  =  0  and  N=10,  S0=N  =  10. 

Formula  (4)  expresses  the  sum  of  1  +  2+3 |-N. 

10(10  +  1)      110 

(2)  Ifm  =  landN=10    S1==         ,/       =—=55. 

Formula  (5)  expresses  the  sum  of  1-  +  2-  +  3- +  N5. 

10  v  11  v  °l 

(3)  If  m  =  2  and  N=10,  S»= ^-^-=385. 

PILING  OF  BALLS  AND  SHELLS. 

342.  Balk  and  shells  are  usually  piled  in  threo  different  forms,  called  trian- 
gular, square,  or  rectangular,  according  as  the  figure  on  which  the  pile  rests 
is  triangular,  square,  or  rectangular. 

(1)  A  triangular  pile  is  formed  by  continued  horizontal  courses  of  balls 
shells  laid  one  above  another,  and  these  courses  or  rows  are  usually  equilateral 
triangles  whose  sides  decrease  by  unity  from  the  bottom  to  the  top  row,  which 
is  composed  simply  of  one  shot. 

Denoting  by  N  the  Dumber  of  balls  contained  in  on  of  the  equilateral 

triangle  which  forms  the  base  of  the  triangular  pile,  it  is  evident  thai  tl  e  num- 
ber of  balls  in  the  base  will  bo  expressed  by  1+2+3  .  .  .  +  N  or  S,  which 
by  (4)  is  equal  to 

N3+N 


PILING  OF  BALLS  AND  SHELLS.  42„ 

If  in  tnis  expression  N  is  successively  replaced  by  the  numbers  1,  2,  3  ...  ., 
the  number  of  balls  in  the  successive  layers,  beginning  at  the  top,  will  be  ob- 
tained.    These  are, 

i»-fi 

in  the  first,  — - — =1 ; 

22+2 
in  the  second,  — - — =3  ; 

32+3 
in  the  third,  — - — =G  ; 

2 

42+4 
in  the  fourth,  — - —  =  10. 

Whence  the  sum  of  the  whole  number  of  balls  contained  in  the  pile  is 

1»+1     22+2     32+3  N2+N 

~ o~~ ^       o     +     7T~  •  •  •  +       o      ' 

iv  <w  &  <w 

which  is  sometimes  used.     A  better  form  may  be  obtained  from  this  by  writing 
it  first 

19_|_22+33  .  .  .  +  N2     1  +  2  +  3  .  .  .  +N 


or 


or 


2^2' 
Si+S!      1/2N3+3N2+N     N2+N\      N3+3N*+2N 


+  S1_1/2N3+3N2+N     N2+N\      ] 
~2~    =2\         ~~ 6  ^~~ 2      /  = 

N(N  +  l)(N  +  2) 


6 
the  most  convenient  expression  for  the  number  of  balls  in  a  triangular  pile 

EXAMPLE. 

How  many  balls  in  a  triangular  pile,  the  side  of  whose  base  contains  35  ? 

35(35+l)(35  +  2) 
Ans.      v     ^  jv     ^     =7770. 
6 

(2)  A  square  pile  is  formed  by  continued  horizontal  courses  of  shot  laid  one 
above  another,  and  these  courses  are  squares  whose  sides  decrease  l>y  unity 
from  the  bottom  to  the  top  row,  which  is  also  composed  simply  of  one  shot ; 
and  hence  the  series  of  balls  composing  a  square  pile  is 

N(N+1)(2N+1) 
1  +  4  +  9  + 1G  +  25-1 |-N2=Sa=  '    -  , 

where  N  denotes  the  number  of  courses  in  a  pile. 

EXAMPLE. 

If  a  side  of  the  base  of  a  quadrangular  pile  contains  35  balls,  how  many  in 

the  pile  ? 

35X3(5X"1 

Ans. s =  14910. 

o 

(3)  A  rectangular  pile  is  one  in  which  the  layers,  except  the  uppermost,  are 
arranged  in  rectangles.  Representing  by  m  +  1  the  number  of  balls  in  the 
top  row,  the  layer  below  it  must  contain  2  rows  of  m+2  balls,  the  next  layer 
3  rows  of  ??i  +  3  balls,  and  so  on,  to  the  N'\  which  contains  N  rows  of  ??j  +  N 
balls  each  ;  and  the  number  in  this  pile  is 


424  ALGEBRA. 

(m+l)+2(m+2)-j-3(i»+3)+4(m+4)+  ....  N(i»+N) 
=m+2j»+3m+4ro+  ....  Nm-4rl9+2  -^9  -f-48+  ....  N- 

=m(l  +  2+34-4+  .  .  .  .N)-f  square  pile 

N(N+1) 

= .  7n-\-  square  pile. 

(4)  The  number  of  balls  in  a  complete  triangular  or  square  pile  must  evi- 
dently depend  on  the  number  of  courses  or  rows;  and  the  number  of  balls  in 
a  complete  rectangular  pile  depends  on  the  number  of  courses,  and  also  on  the 
number  of  shot  in  the  top  row,  or  the  amount  of  shot  in  the  latter  pile  depends 
on  the  length  and  breadth  of  the  bottom  row;  for  the  number  of  courses  is 
equal  to  the  number  of  shot  in  the  breadth  of  the  bottom  row  of  the  pile. 
Therefore,  the  number  of  shot  in  a  triangular  or  square  pile  is  a  function  of  N, 
and  the  number  of  shot  in  a  rectangular  pile  is  a  function  of  N  and  m. 

The  expression  for  a  rectangular  pile, 

N(N+1)        N(N  +  1)(2N  +  1) 
__ m+ _ f 

may  be  written 

N(N  +  l)(3/n  +  2N+l)      1 

6  ;=gN(N+l)[2(m+N)+m+lJ. 

But  m  +  1  is  the  number  of  balls  in  the  top  row,  N  is  the  number  in  the  smaller 
side  of  the  base,  and  ra-j-N  the  number  in  the  greater  side,  2(m-f-N)  the 

L      •     ,                    „  ,                  .,                          N(N-fl) 
number  in  the  two  parallel  greater  sides ;  moreover, is  the  number 

of  balls  in  the  triangular  face  of  each  pile;  hence  we  have  also  this  general 
-ule  for  rectangular  or  square  piles. 

RULE. 

Add  to  the  number  of  balls  or  shells  in  the  top  row  the  numbers  in  its  two 
parallels  at  bottom,  and  the  sum  multiplied  by  one  third  of  the  slant  end  or 
face  gives  the  number  of  balls  in  the  pile. 

EXAMPLES. 

(1)  How  many  balls  are  in  a  triangular  pile  of  15  courses  ?         Ans.  6&0- 

(2)  A  complete  square  pile  has  14  courses:  how  many  balls  are  in  thf»  pile, 
and  how  many  remain  after  the  removal  of  5  courses  ?      Ans.  609  and  »54. 

(3)  In  an  incomplete  rectangular  pile,  the  length  and  breadth  at  bottom  are 
respectively  4G  and  20,  and  the  length  and  breadth  at  top  are  35  and  9  -.  how 
many  balls  does  it  contain  ?  Ans.  71 

(4)  The  number  of  balls  in  an  incomplete  square  pile  is  equal  to  fi  times 
the  number  removed,  and  tho  number  of  courses  left  is  equal  to  tho  number 
of  courses  taken  away  :  how  many  balls  were  in  the  complete  pilo  ? 

Ans.  385. 
(.'))   Let  h  and  k  denote  the  length  and   breadth  at   top  of  a   recUngubu 
truncated  pile,  and  N  tho  number  of  balls  in  each  of  the  Blanting  edges;  tLen. 
if  B  bo  tho  number  of  balls  in  the  truncated  pile,  prove  that 

N<  ) 

B=-  \  2N9+3N(A+Jfc)+6Mr— Stt+fc+NJ+l  £ . 


VARIATION.  425 


VARIATION. 

343.  Let  a   Jenote  a  constant  quantity,  or  one  which  does  n.  t  change  its 
talue,  and  x  a  variable  which  is  supposed  to  increase  or  diminish. 

The  product  of  the  quantities  a  and  x  being  denoted  by  X,  if  x  is  increased 

or  diminished,  X  will  be  increased  or  diminished  in  the  same   proportion. 

Thus,  if  x  become  x',  and,  consequently,  X  become  X',  we  shall  have 

x:x'  ::X:X', 

for 

ax      x      X 
ax=X  and  ax'=X'  .-.  — ,=— =tt„  or  x  :  x' : :  X  :  X'. 

(IX        x        Ji. 

Under  these  circumstances  X  is  said  to  vary  directly  as  x. 

The  symbol  of  variation  is  x  ;   and  the  expression  X  varies  directly  as  x,  is 
indicated  by  the  combination  of  symbols  X  <x  x. 

344.  If  the  product  of  x  and  y  be  constant,  and  x,  y  both  variable,  since 

xyz=x'y'  =  C, 

1      1 

J  J  y         y< 

In  this  case  as  x  varies  as  the  reciprocal  of  y,  x  is  said  to  vary  inversely  as  y, 
and  the  symbolical  expression  is 

1 

X  OC   -. 

y 

If  xy=X  and  x'y'  =  X',  then  X  :  X'  : :  xy  :  x'y'. 

The  variation  of  X  in  this  case  depends  on  the  variation  of  two  quantities 
V  and  y,  which  is  expressed  thus, 

X  oc  xy. 

X  X'  XX' 

345.  If  xy=X  and  x'y'*=X',  then,  *=—  and  x'=—  .-.  .r  :  x?  :  :  — -  :  — -. 

In  this  case  x  is  said  to  vary  as  X  directly,  and  as  y  inversely.     The  symbol  is 

X 

x  oc  — . 

y 

x     y  y      z 

346.  Letxx2/,i.e.,.r:.r'::2/:2/'or-='-,andlet?yo:2,i.e.,y  y' :  z : : z' or  -  =r- 

•i     y  y     z 

X       z 

X'        2 

Gnat  is,  if  one  quantity  vary  as  a  second  and  the  second  as  a  third,  the  first 
varies  as  the  third. 

1  1 

347.  In  like  manner,  if  .t  qc  y  and  y  oc  -,  x  oc  -. 


— =—  or  x :  x' : :  z  :  z',  i.  e.,  xoc  z  ; 


2 


Again,  let  xoc  y  and  zoc  y  .'.  xoc  z,  or  x:x'::z:z',  or  x:z::x':2' 

.-.  x±2:z::x'±z'  :z',  or  x^zz:x'±z' 
But  z:z'  ::y:y',  .-.x^zZix'^zz'-.-.y-.y',  l.  e.,  yxx±z. 

Again,  since  xocy,  x:x'  ::y  :  y',  and  since  zccy,  z  :;'::;/ :?/',  .-•  xz:.r"2 
.  y- :  y'2,  and  tfxz  :  yftfz' : :  y :  y',  or  y  a  -y/1-  »  tnat  is>  ^  Uvo  quantities  vary 
respectively  as  a  third,  their  sum,  difference,  or  pquare  root  of  their  product, 
varies  as  this  third  quantity. 

348.  If  x  x  y  and  m  be  a  constant  quantity,  integer  or  fractional,  since  x :  y  : : 


42G  ALGEBRA. 

x' :  y', .-.  x :  y  : :  mx' :  7ny'  (Art.  107),  i.  e.,  x  oc  my  ;  that  is,  if  one  quantity  vary 
as  another,  it  varies  as  any  multiple  or  part  of  this  other. 

When  x  cc  y,  and,  consequently,  x  oc  my,  so  that  x  :  x' : :  my  :  my'  or  x  :  my 
::x':my',  then,  if  x=?ny,  x'  will  be  equal  to  my'  in  all  cases;  whence,  if  .r 
vary  as  y,  x  is  equal  to  y  multiplied  by  some  constant  quantity. 

349.  If  X  and  Y  are  two  corresponding  values  of  x,  y, 

X=mY,  .-.  7n  =  y:', 

from  which  it  follows  that,  when  two  corresponding  values  of  x,  y  are  known, 
the  constant  m  may  be  found. 

350.  Let     xx  y  .-.  x  :  x' :  :y  :y' .•.  xm  :x'm  :  :ym  :y'm  .•.  xma:ym; 

m  being  any  exponent  integer  or  fractional.  Whence,  if  one  quantity  vary  as 
another,  any  power  or  root  of  the  first  quantity  will  vary  as  the  same  power 
or  root  of  the  second  quantity. 

351.  Let  xij,  and  let  t  be  another  quantity,  either  variable  or  constant,  and 
of  which  I,  I'  are  either  equal  or  different  values.     Then,  since 

xazy,  x :  x' : :  y  :  y\  and  t:t'::t:t'; 
.•.  xt :  x't' :  :yt:  y'l',  or  xtccyt ; 
x   x'      y   ?/'        x    y 

Vl'-'-'l'-'l"01"^!' 

that  is,  if  one  quantity  vary  as  another,  and  if  each  of  them  be  multiplied  or 
divided  by  any  quantity,  variable  or  constant,  the  products  or  quotients  will 
vary  as  each  other. 

XV  x 

Consequently,  if  xoc  y,  -a  -,  or  —  oc  1. 

x 
Whence,  i(  x  <x  y,  -  is  constant. 

J  y 

352.  Let  xy  a  X,  i.  e.,  xy  :  x'y' : :  X  :  X  ; 
by  alternation,  xy  :  X  : :  x'y' :  X' ; 

X  X'  X 

and  similarly,  .roc—  ; 

y 

that  is,  if  the  product  of  two  quantities  vary  ns  a  third  quantity,  each  of  the 
two  quantities  varies  as  the  third  directly,  ami  as  the  other  inversely. 

353.  If  X=X'=  constant,  xy :  1 : :  x'y' :  1 ; 

1  1  1 

.-.  .r :  -  : :  x' :  —  ,  or  x  x  -  ,- 

y        y  y 

that  is,  if  the  product  of  two  variable  quantities  be  constant,  these  quantities 
vary  inversely  as  each  other. 

354.  Let  a  be  a  constant,  and  .r,  y,  z  variables,  and  let 

a  :  x  : :  y  :  z,  a  :  x' : :  y' :  :  .  A 
.-.  az=.ry,  az'  =  .i'i/',  &c. ; 
.-.  az  :  (/:' :  :  xy  :  .r'//',  or  :::'::  xy  :  x'y' 
.-.  z  x.ry  ; 

that  is,  if  four  quantities  arc  always  proportional,  and  one  or  two  of  them  are 
constant,  tl thers  being  variable,  it  can  be  found  bow  the  latter  vary. 

355.  Let  x,  y,  z  be  three  quantities,  of  which,  c  cy  when  :  is  constant,  and 


SYMMETRICAL  FUNCTIONS.  427 

x<xz  when  y  is  constant;  it  is  required  to  determine  the  variation  of  X  wheu 
y,  z  are  both  variable. 

Suppose,  first,  that  .r  is  made  to  vary  as  y,  and  that  wheu  y  becomes  y',  x 
becomes  x'. 

Next,  that  x'  (varied  from  x  by  the  variation  of  y)  is  made  further  to  vary 
as  z,  and  that  when  z  becomes  z',  x'  becomes  x".     Then,  since 

x  :  x' : :  y  :  y' ,  and  x' :  x"  :z:z' 
.•.  xx' :  x'x"  ::yz:  y'z', 
or  x :  x"  ::yz:  y'z' ;   ' 

i.  e.,  x<xyz. 
Therefore,  if  x  vary  as  y  when  z  is  constant,  and  as  z  when  y  is  constant, 
when  y,  z  are  both  variable,  x  varies  as  the  product  yz. 

Similarly,  it  can  be  proved,  that  if  t  vary  as  v,  x,  y,  z  separately,  the  others 
being  constant  when  v,  x,  y,  z  are  all  variable,  t  varies  as  the  product  vxyz. 


SYMMETRICAL  FUNCTIONS  OF  THE  ROOTS  OF  AN  EQUA- 
TION. 

356.  There  are  certain  functions  of  the  roots  of  an  equation  which  may  be 
expressed,  in  a  general  manner,  by  means  of  the  coefficients  of  that  equation, 
without  the  equation  itself  being  resolved. 

These  functions,  which  form  a  very  extensive  class,  are  termed  rational 
and  symmetric  functions,  or  simply  symmetric  functions. 

They  are  called  rational,  because  the  roots  do  not  enter  into  them  under 
the  radical  sign,  nor  with  fractional  exponents;  the  roots  are  combined  only 
by  addition,  subtraction,  multiplication,  and  division.  These  functions  are 
called  symmetric,  because  the  roots  are  combined  in  such  a  way  that  any  two 
of  them  may  be  interchanged  without  altering  the  value  of  the  function. 

For  example,  the  expressions 

ab      ac       be 
ac+bc+ab,  a°-+b'+c\  — +  — +  — _3a6c 

are  rational  and  symmetric  functions  of  a,  b,  c. 

All  the  coefficients  of  an  equation  are  symmetric  functions  of  its  roots,  as 
may  be  seen  in  the  expressions  for  the  coefficients  in  Art.  245  ;  for,  in  these 
expressions,  if  at  were  written  in  every  place  where  a2  occurs,  instead  of  a2, 
and  a2  in  every  place  where  <?!  occurs,  instead  of  aM  or  if  any  other  two  of 
the  roots  were  interchanged,  the  values  of  the  expressions  would  not  be 
altered. 

Several  quantities,  a,  b,  c,  &c,  being  given,  if  we  arrange  them  two  and 
two,  in  every  possible  way,  and  if  in  each  arrangement,  e.  g.,  ab,  we  give  the 
exponent  a  to  the  first  factor  and  the  exponent  0  to  the  second,  we  have  a  se- 
ries of  products  such  as  a"bP,  whose  sum  is  evidently  a  symmetric  function 
of  the  quantities  a,  b,  c,  &c.  This  function  is  called  a  double  function,  be- 
cause each  term  contains  two  of  the  given  quantities  ;  it  is  represented, 
abridged,  by  S(aab@),  the  letter  S  being  here  employed  to  denote  the  word 
sum.  In  like  manner,  triple,  quadruple,  &c,  symmetric  fuuetions  are  repre- 
sented by  S(a"ZA>),  S(a"b^ds),  &c. 

In  accordance  with  this  notation,  simple  symmetric  functions,  as  aa-\-ba 


428  ALGEBRA. 

-\-ca-\- ,  will  be  represented  by  S(a"),  which,  for  the  sake  of  abridgu.ent, 

is  ordinarily  written  Sa.     In  like  manner,  we  have 

<l  =  a  +b  +c  +  ... 

&c.  cVc. 

The  notation  of  which  we  have  been  speaking  applies  to  entire  symmetric 
functions;  but  when  the  terms  of  a  symmetric  function  are  fractional,  we 
can,  by  reducing  them  to  a  common  denominator,  express  the  function  by  a 
single  fraction,  whose  numerator  and  denominator  are  integral  symmetric 
functions.     Thus : 

ab       ac       be 

which  is  a  fractional  symmetric  function  of  a,  b,  c,  becomes,  by  reduction, 

a*b* + aV + b*c* — 6a*b*c* 

357.  An  equation  being  given,  to  find  the  sums  Si,  S2,  &c,  of  the  like  and 
entire  forcers  of  its  roots. 

Let  the  equation  be  X  =  0, 
or  xm4-Pxra-1  +  Q.rm-3+R.rm-3  .  .  .  +Tx+U  =  0  ....  (1) 

and  call  the  m  roots  a,  b,  c,  d. 

We  can  find  by  Art.  238  the  quotients  obtained  by  dividing  X  by  each  of  its 
factors,  x — a,  x — b,  x — c,  &c. ;  and  we  know  (Art.  253)  that  by  adding  these 
m  quotients  together,  the  sum  must  be  equal  to  the  derived  polynomial  X',  or 

m.r™-i+(wi— l)P;rn'-2  +  («i  —  2)Q.cm-34-(wi— 3)Rxra-^.  .  .  -f  T. 
The  coefficients,  therefore,  of  the  powers  of  .t,  in  this  sum,  must  be  equal  to 
the  coefficients  of  the  same  powers  of  x  in  the  derived  polynomial  X',  each  to 
each.     In  this  manner  the  required  sums  can  be  determined. 
Let  us  take,  then,  the  quotient  of  X  divided  by  .r — a, 
X 


_.rm-1-r-«  .rm-2+a2 

x (l 

-fP|         +Pa 

+  Q 


rm-3_j_a3 

+  Pa- 
+  Qa 
+  11 


;m— *  .  .  .  -fa"1-1 
+  Pam-9 
4-Qa™-3 


+  T. 
In  order  to  have  the  other  quotients,  it  will  bo  sufficient  simply  to  substitute 
for  a,  in  this  expression,  successively  b,  c,  d,  &c.      [f  we  add  these  quotients, 
and  put  Si,  Sa,  S3,  &c,  instead  of  the  sums  a-\-b-{- c-{-  .  .  .,  a9+6*+C*-f"  •  • 
a3 -J- 6s -J- c3-}-  .  . .,  we  shall  have 
mxn 


n-l+s, 

.rm--+S, 

X"n-»+S3     .r"1-*.. 

•  •    -\-  ^m— 1 

+  mP 

+ps, 

+PS 

+  PSm_* 

+  '"(i 

+QS, 

+QSm^ 

+  ///K 

+  RSm_, 

+  »iT. 
Hence,  equating  the  coefficients  of  corresponding  terms  :n  these    denfiral 
repressions,  we  get 


SYMMETRICAL  FUNCTIONS.  42<» 

Si+mP—(m— 1)P, 

S2+PS,  +  mQ=(m— 2)Q, 
S3+PS,  +  QS1+mR=(m— 3)R, 


S^  +  PS^+QS^ |-mT=T, 

or,  simplifying, 

S1  +  P  =  0, 

S2+PS1  +  2Q=0, 

S3+PS*+QSi  +  3R  =  0,  (2) 


Sm_i+PSm_2+QSm_3  •  .  •  +(m-l)T=0. 
By  means  of  these  equations  it  will  bo  easy  to  calculate  successively  Si,  S2, 
S3,  &c,  and,  finally,  Sm_ 1,  i.  e.,  the  sums  of  all  the  similar  powers  of  the  roots 
whose  index  is  less  than  the  degree  of  tho  equation.     In  order  to  determine 
the  sums  of  tho  higher  powers,  expressed  by  S,n,  Sm+i,  Sm+2,  &c,  Ave  substi 
rute  successively  a,  b,  c,  .  .  .  in  equation  (1),  and  thus  obtain 

am_|_pam-l_^.Qam-3 _|_T«  +  U  =  0 

i">4.Pin,-l^_Qim-3 +T6+U=0 

&c. 
Wo  multiply  these  m  equalities  respectively  by  an,  b",  &c,  and  then  add 
them ;  we  thus  obtain 

Sm+n-f"PSm+n— l  +  QStn+n— 2  •   •  •  •    -f-  TSll+1-{-USn  =  0. 

We  can  make  successively  «=0,  1,  2,  &c.,  and  thus  determine  Sm,  Sm+i, 

Sm+2, ;  we  find 

Sm    +PSm_1+QSm_3 .  .  .  +TS1+US0=0 
Sm+1+PSm    +QSm_1...  +TS.,+US,  =  0  (3) 

Sra+2+PSm+,  +QSra  .  .  .  +  TS3+USa=0 
In  the  first  of  these  equations  we  can  put  in  place  of  US0,  mil,  for  S0 
=a°-r-o°-4-c°-{-  .  .  .  =m;  we  shall  thus  find  that  these  formulas  follow  the 
same  law  with  those  in  (2).  By  means  of  the  first  of  theso  we  can  determine 
Sm,  and,  passing  successively  to  each  of  the  succeeding  formulas,  we  shall  bo 
able  to  determine  each  new  sum  by  means  of  the  sums  already  calculated. 

It  may  be  well  to  observe  that  all  the  sums,  Si,  S2,  S3,  &c,  may  be  ex- 
pressed without  any  denominator  in  functions  of  P,  Q,  R,  &c.  This  results 
from  the  fact  that  the  first  term  in  each  of  the  relations  (2)  and  (3)  has  unity 
for  its  coefficient. 

EXAMPLES. 

(1)  For  a  numerical  application  take  the  equation  x3 — 7.r 4-7=0.  Here 
P  =  0,  Q=—  7,  R=7.  Since  P=0,  the  relation  S,  +  P=0  gives  S1=0. 
Tho  relations,  then,  which  determine  the  sums  Si,  S2, . . .  S6,  reduce  them- 
selves to 

S,=0,  S.2+2Q=0,  S3+3R  =  0, 
S4+QS2=0,  S6+QS3+RSs=0,  S6-fQS4+RS3=0; 
and,  by  substituting  the  values  of  Q  and  R,  we  readily  find 

S,=0,  S2=14,  S3=— 21,  S4=98,  S6=— 245,  S6=833. 

(2)  Calculate  the  sums  of  the  similar  and  entire  powers  of  the  rooi-s  of  th 
equation  x4 — .r3— 19x2+49x— 30=0. 

Ans.  S,  =  l,  Ss=39,  S3=  -89,  S4=723,  S5=— 2849,  S6=16419,  &c 


430  ALGEBRX. 

(3)  t«+rr+«=0 

Ads.  S,  =  0,  S.  =  0,  SJ=—3r,  S4= — is,  Ss=0,  SgsSf*. 

:358.  In  the  equation  Sm+n+PSID+1,_^-QS„H.II_9 |-TSt+1  +  USn=0 

n  can  be  a  negative  number,  and  thus  the  sums  of  the  negative  powers  of  the 

roots  can  be  determined.     But  it  will  be  more  simple  to  change  x  into  -  in 

the  proposed  equation,  and  to  find  successively,  by  means  of  formulas  (2)  and 
(3),  the  sums  of  the  positive  powers  of  the  roots  of  the  transformed  equation. 
It  is  evident  that  these  powers  are  the  negative  powers  of  a,  b,  c,  ... . 

359.   To  determine  double,  triple,  Sec.,  functions,  represented  by  S(aab^), 
S(aazA-y),  &c. 

In  order  to  find  S(a"£r)  we  multiply  together  the  two  sums 

aa+ba+ca-\ =S«, 

we  have 

+  «"^  +aV34-iV?H 

This  product  contains  two  series  of  terms.  The  first  series  is  the  sum  of  all 
the  powers  a-{-/3  of  the  roots,  and  may  bo  expressed  by  Sa_j-/3 ;  the  second 
series  is  the  sum  of  all  the  products  which  are  formed  by  multiplying  the 
power  a  of  any  root  whatsoever  by  the  power  ft  of  any  other  root,  and  may 
be  expressed  by  S(a"bft).     We  have,  then, 

Sa+/?+S(a"6^)  =  SaS,;: 
and  from  this  equation  we  derive,  for  double  functions,  the  formula 

S(aa6^  =  SaS/3-Su+/?. 
To  find  the  triple  function  S{aab^cy),  multiply  together  the  three  sums 

<Ia+fca+ca-f-...  =  Su, 

a«+bS+cV+...±=Sp, 
ay+bY+cr  +  ...  =  SY. 
The  product  is  a  symmetric  function,  which  evidently  comprises  all  the 
terms  contained  in  each  of  the  five  forms 

aa+P+r,  aa+/V,  aa+*b0,  a^'b\  a%licy ; 
nence  we  have 

SaW,+S(aa+/V)  +  S(an+>^)  > 

+  S(/+>7/')  +  S(aai'V)  p^b>" 
But  the  formula  for  double  functions  gives 

S(aa+V)=Sa+j8Sy--SB+J8+y> 

S(aa+> b0)  =  Sa+ySp-  Su+/Hr, 

S(/+)'«a)  =  S^+>Su-S,i+.+;. 

By  substituting  these  values  in  the  preceding  equality,  and  then  deriving 

from  this  equality  the  value  of  S(</'7.  V"),  we  obtain  for  triplo  functions  the 

formula 

S(an^cy)  =  SaS/jSy-Sa+^Sy-Sa4)S,(_S,H)SiI  +  -jS,:;,+>.. 

In  tho  same  manner  might  the  quadruple  function  S{a"lrcYcr),  or  the  sum 
of  any  succeeding  combinations,  bo  expressed  by  tho  stuns  of  the  powers. 


SYMMETRIC  FUNCTIONS.  431 

360.  Every  rational  and  symmetric  algebraic  junction  of  the  roots  of  an 
equation  can  be  expressed  rationally  by  the  coefficients  of  that  equation. 

Since  Si,  S3,  S3,  &c,  can  bo  expressed  without  denominators  (Art.  357)  in 
functions  of  the  coefficients  of  the  proposed  equation,  and  the  double,  triple, 
quadruple,  &c,  functions  can  be  expressed  by  the  sums  of  the  powers,  it  fol- 
lows that  all  these  symmetrical  functions  can  be  expressed  by  integnd  func- 
tions of  the  coefficients.  And  as  every  symmetrical  polynomial  in  a,  b,  c . . . 
must  be  composed  of  the  assemblage,  by  addition  or  subtraction,  of  several 
symmetric  functions  of  the  form  S{aab^cYd&  . . .),  it  follows  that  the  value  of 
every  rational  symmetric  function  whatever  of  the  roots  of  an  equation  (with- 
out the  roots  being  known)  can  bo  expressed  by  the  coefficients  of  the  equa- 
tion. 

USE  OF  SYMMETRIC  FUNCTIONS  IN  THE  TRANSFORMATION  OF  EQUA 

TIONS. 

361.  Symmetric  functions  present  themselves  in  the  transformation  of 
equations,  whenever  the  roots  of  the  transformed  equation  must  be  rational 
functions  of  the  roots  of  the  given  equation. 

Let  a,  b,  c  ...  be  the  roots  of  the  given  equation ;  for  the  sake  of  definite- 
ness,  I  suppose  that  two  of  its  roots  enter  into  the  composition  of  each  root 
of  the  transformed  equation,  and  I  represent  by  F(a,  b)  the  rational  function 
which  expresses  the  law  of  this  composition. 

Suppose  that,  after  we  have  made  all  these  combinations,  two  and  two,  of 
a,b,c...  we  put  successively  in  F(a,  b)  instead  of  a  and  b,  the  two  roots  of 
each  arrangement,  it  is  clear  that  wo  shall  thus  have  all  the  roots  of  the  trans- 
foi-med  equation,  to  wit : 

F(a,  b),  F(a,  c), F(6,  a),  F{b,  c) &c 

Consequently,  this  equation,  decomposed  into  factors,  will  be 
[z-F(a,  b)]  [z—F{a,  c)]  .  .  .  .  =0. 

This  product  does  not  vaiy  in  making  between  a,  b,  c  .  .  .  .  the  proposed  ex- 
change ;  for,  if  we  make  the  change,  the  factors  can  only  place  themselves  in 
some  other  order.  We  are  sure,  then,  that,  after  the  multiplication,  the  co- 
efficients of  the  different  powers  of  2  will  be  symmetric  and  rational  functions 
of  a,  b,  e  .  .  . 

Thus,  by  following  the  method  of  procedure  hitherto  explained,  we  can 
express  these  coefficients  by  means  of  those  of  the  proposed  equation. 

362.  But  there  exists  another  method,  often  preferable,  of  employing  sym- 
metric functions. 

It  is  founded  on  the  observation  that  the  relations  [2]  and  [3]  in  Art.  357, 
existing  between  the  coefficients  of  an  equation  and  the  sums  of  the  similar 
powers  of  its  roots,  can  be  used  to  discover  the  coefficients  of  the  equation 
when  they  are  unknown,  provided  wo  know  these  sums  as  far  as  that  sum  of 
the  powers  whose  order  is  equal  to  the  number  of  unknown  coefficients,-  i.  e., 
to  the  degree  of  the  equation. 

Hence,  to  arrive  at  the  transformed  equation,  we  determine,  first,  of  what 
degree  this  equation  is  to  be.  Wo  next  find  the  sums  of  the  first,  second,  &c., 
powers  of  its  roots,  as  far  as  the  sum  of  the  powers  whose  order  is  equal  to 
the  degree  of  this  transformed  equation;  then,  by  means  of  these  sums,  we 
calculate  the  unknown  coefficients.     It  is  clear  that  these  different  sums  are 


432  ALGEBRA. 

symmetric  functions  of  the  roots  of  the  proposed  equation,  and  that  they  can 
be  expressed  by  the  coefficients  of  this  equation.  Hence  they  can  readily  be 
determined. 

3G3.  As  an  illustration  of  the  preceding  method,  I  will  resume  here  tin- 
question  of  the  equation  of  the  squares  of  the  dilferences,  already  treated  of 
in  Art.  278.  Symmetric  functions  give  the  most  simple  and  elegant  solution 
of  which  it  is  susceptible.     The  question  is  this  : 

To  find  the  equation  whose  roots  are  the  squares  of  the  differences  of  the 
roots  of  a  given  equation, 

xra+Pxra-I  +  Qxm-I4-.  .  .  .  =  0 [A] 

Represent  the  transformed  equation  by 

zn4-jpzn-1+92I1-*4-rz,,-8+.  .  .  .  +  ?r  +  w  =  0  .  .  .  [B] 

The  m  roots  of  [A]  being  a,  b,  c  .  .  .  those  of  [B]  will  be 

{a— by,  (a—c)°,  (a  —  d)-,  .  .  .  (6— c)3,  .  .  .  {b—dy,  {c—dy,  .  .  .  &c 

The  number  of  these  squares  is  evidently  that  of  the  combinations,  two  and 
two,  that  can  be  made  with  the  m  quantities,  a,  b,  c  .  .  .  ;  hence  the  degree  of 
the  required  transformed  equation  will  be  n=\m{m  —  1). 

The  coefficients  p,  q,  r  .  .  .  may  easily  be  found  when  we  know  the  sums 
of  the  similar  and  entire  powers  of  the  roots  of  equation  [B]  ;  since  the  sum 
of  the  first  powers  is  equal  to  that  of  the  nlh  powers.  Let  us  designate  these 
new  sums,  then,  by  f,f2,f3,  <fcc,  and  find  the  general  value  of/a,  a  being  any 
entire  and  positive  number  whatsoever. 

The  roots  of  the  equation  [B]  are,  as  has  already  been  stated,  (a  —  by,  cVc- 
Raising  these  roots,  then,  to  the  power  a,  we  have 

/a=(a— 6)M+(a— cyt+ia— dfa [■{b-c)-i+,  &c. 

In  order  to  find  this  sum,  consider  tho  expression 

<p{x)  =  {x— aya+(x— b)"a  +  {x— c)ia+ 

which  contains  the  m  binomials  .r — a,  x — b,  x — c If  we  make  in  this 

expression  successively  x=a,  b,  c,  .  .  .,  and  add  the  m  results,  we  evidently 

obtain 

2fa=<p(a)  +  <p(b)  +  $(c)  +  .  .  . 

If  we  develop  the  powers  which  compose  tf>(.r),  we  find 

2a(2a— 1) 

x2a — 2aax-a~l  +  ,t — ia»lBa-«+ asa 

6(r)=i  2n(2a  —  1) 

+,  <fec. 
or,  more  simply,  by  using  tho  notation  Slt  S.:.  Arc, 

2o(2a— .n 

#(x)  =  m.i--,-2aSIiaa-,  +  — -  — —  ...  +  SM. 

Substituting  a,  b,  r,  .  .  .  in  this  expression  instead  of  x,  and  adding  tho  re- 
sults, we  obtain 

2n(2a  — 1) 

-,;/;,  =  ,»S.,,--.'..S,S.,I„1  +      \2     ;S,S,a_2  .  •  •  +>' 

In  this  second  member  it  will  be  perceived  thai  the  terms  at  an  equal  dis- 
tance from  the  extremes  are  equal;  consequently,  stopping  at  the  middle  term 

of  the  expression,  and  taking  only  the  hall' of  that  term,  we  have  the  geMnl 
vn'-ie  of_/;,,  to  wit, 


QUADRATIC  FACTORS  OF  EQUATIONS.  433 


,/u — mh2a  —  2abiSja_i-j - — — — S2SJU_2 .  • . , 


5a(2a  — 1)( 
1.2 


I    |2«(2a-l)(2«-L)...(a+l)c, 
2  1  .  (>    3         a  ba^a- 

As  the  signs  are  alternately  -{-  and  — ,  there  will  never  be  any  uncertainty 
as  regards  this  last  term.  Let  us  view,  then,  the  operations  which  must  be 
performed. 

1°.  "We  calculate  the  sums  Si,  S3,  S3..  up  to  S:a  by  means  of  the  known 
relations  S,  +  P=0,  S2+PS!  +  2Q=0,  &c. 

2°.  In  the  formula  which  expresses^  we  make  successively  a=l,  2,  3, 
..«,  and  we  thus  have,  to  determine  the  n  Bums fuf2,f3,  ...fn, 
/=mS.2  — S1S„/,=mS4— 4SiS3+3S:S2,  &c. 

3°.  Finally,  the  relations  existing  between  these  n  sums  and  the  n  coeffi 
cients^j,  q,  r,  ...  will  give  the  values  of  these  coefficients,  viz., 

P=~fu  q=-h(fi+PAh  r=-l(f,+  pfi+gft),  &c. 

364.  A  method  entirely  analogous  to  that  which  has  been  employed  in  find 
ing  the  equation  of  the  squares  of  the  differences  can  be  employed  in  a  great 
number  of  cases,  and  particularly  in  those  where  the  roots  of  tho  transformed 
equation  are  similar,  and  entire  powers  of  the  difference,  of  the  sum,  of  the 
product,  or  of  the  quotient  of  any  two  roots  whatsoever  of  the  given  equation. 

For  example,  suppose  that  each  new  root  is  to  be  the  power  k  of  the  sum 
a-\-b  of  two  roots  of  equation  [A].  Taking  ns=|m(m  —  1),  the  transformed 
equation  ought  to  have  the  form 

2n_J_p2n-l.J_g2n-3_|_ [-tZ-\-U  =  Q   .....    [C] 

and  if  we  make 

fa  =  (a+b)^+(a  +  cY«+  . . .  +(6+c)k«+,  &c, 
the  calculation  will  reduce  itself  to  expressing/a  by  a  general  formula.     To 
do  this,  we  take  the  function 

^(x)  =  (x+a)"«+(.r+&)^+(x+c)ka  +  ,&c, 
the  development  of  which  is 

kaUca  —  1) 
^(a:)=maJlg+feaSia*n-1+  '  &&**-*+  . .  .  +  Ska. 

J.       •      Urn/ 

But  if,  before  the  development,  we  substitute  in  ^(.r)  successively  a,  b,  c, 

.  .,  instead  of  x,  the  sum  of  the  resultants  will  be  equal  to  ~fa-\- 2kaSk„; 

hence  it  is  easy  to  perceive  that  by  making  the  same  substitutions  in  the 

development,  we  shall  have 

2/a+2kflSta=mSk,+^aS1Ska_,  . . .  +mSk«. 

Finally,  we  derive  from  this  equation  the  required  formula, 

kalka  —  1) 
/a=(m-2ka-1)Ska+7caS1Sk,_1+-Ar-^ !S,Sku_.2+,  &c. 

When  ka  is  even,  we  stop  at  the  term  which  contains  S  with  two  equal  in- 
dices, and  we  take  only  the  half  of  it ;  but  when  ka  is  uneven,  we  stop  at  the 
term  in  which  the  two  indices  are  l(ka  —  1)  and  r,(7ia4-l),  and  we  take  the 
entire  term.  ^ 

QUADRATIC  FACTORS  OF  EQUATIONS. 

365.  Every  equation  of  an  even  degree  has  at  least  one  real  quadratic  factor. 
Let  the  proposed  equation  be 

E  E 


434  ALGEBRA. 

xn-\-plxa'  1+^Jxn_2+ kP°=°»  having  roots  a,  b,  c,  &c,  and  let  ji  =  2/j.  /» 

being  an  odd  number.     Let  it  be   transformed  (Art.  3G2)  into   an   equation 
whose  roots  are  the  combinations  of  every  two  of  its  roots,  of  the  form  »,  =a 
-\-b-\-mab,  m  being  any   number;    also,  let   the    transformed    equation   be 
(£m(?/)  =  0;  then  its  coefficients  will  be  symmetrical  functions  of  a,  b 
and,  therefore,  rational  and  known  functions  of  pu  p:,  ice.  ;  and  its  degree  wiU 

be .which  is  odd;  therefore,  6m(y)=z0  will  have  at  least ;one  real  roou, 

whatever  be  the  value  of  m.  Hence,  making  m=\,  2,  3,  .  . .  {/*('-." — 1)  +  1  \> 
successively,  each  of  the  equations  (p{(y)  =  0,  6:(y)  =  0,  &c,  will  hire  at  least 
one  real  root;  that  is,  we  shall  have  p(2fi — 1)+1  real  values  for  combinations 
of  two  roots  of  the  proposed  equation,  of  the  form  a-{-b-{-mab  ;  but  there  are 
oi\]y  u(2u  —  1)  such  combinations  which  are  differently  composed  of  the  roots 
a,  /;,  c,  cVc. ;  therefore,  two  of  these  combinations,  for  which  we  have  obtain- 
ed real  values,  must  involve  the  same  pair  of  the  quantities  n,  b.  c,  &c. :  I»*t 
this  pair  of  roots  be  a,  b,  and  a,  a',  the  real  roots  of  the  corresponding  equa- 
tions 6jy)=0,  </>m.(t/)  =  0,  so  that 

a-\-b-{-7nab=a,  a-\-b-{-m'ab  =  a  ; 
therefore,  a-\-b  and  ab  are  real,  and  the  proposed  equation  has  at  least  one 
real  quadratic  factor,  and  two  roots,  either  real,  or  of  the  form  o±^V — *• 
Hence  every  equation  whose  degree  is  only  once  divisible  by  2  has  at  least 
one  real  quadratic  factor. 

We  shall  now  prove  that  if  it  be  true  that  every  equation  has  at  least  one 
real  quadratic  factor  when  its  degree  is  r  times  divisible  by  2,  or  when  n=2r/i, 
where  n  is  odd,  the  same  is  true  when  the  degree  of  the  equation  is  r-f-1 
times  divisible  by  2.  For,  let  n=2r+1fi;  then  the  degree  of  the  transformed 
equation  will  be  2r/u(2r+V — 1)>  which  is  only  r  times  divisible  by  2  ;  therefore, 
by  supposition,  the  transformed  equation,  6m(y)  =  0,  will  have  two  roots,  either 
real  or  imaginary.  If  they  are  real,  then,  exactly  in  the  same  way  as  for  the 
preceding  case  of  the  index  being  only  once  divisible  by  2,  it  may  be  shown 
that  die  proposed  equation  has  at  least  one  real  quadratic  factor.  If  they  are 
imaginary,  we  shall  have  y=a±l3-</ —  1,  each  of  which  expresses  the  value 
of  some  one  of  the  combinations  a -\-b-\- »iul>,  a-\-c-\-mac,  &c.  Suppose, 
therefore,  that  we  have  a-\-b-\-mab=a-\- 3-/ — 1  ;  then,  as  shown  above,  we 
can  give  m  such  a  value  m',  that  $m.(y)  =  0  shall  havo  a  root  corresponding  to 
the  combination  of  the  same  letters,  so  that  a-\-b  +  m  ab=a '-\-;i'  V  —  1  ;  from 
which  equations  we  can  obtain  values  of  rib  and  a-\-b  under  the  forms 

a+b  =  y+t  -/^T, 

a6  =  ?'  +  <<V-l, 

...  a*— (y+oV—  lJx-f.y'-l-d'V  —  l  is  a  factor  of /(z) ; 

but  if  any  real  expression  have  a  factor  of  the  form  M-J-N  \/  —  1,  it  most  a.»u 
have  one  of  the  form  M — N  -J  —  1  ; 

...  xi  —  {y  —  dJ~^-i)r+y  —  i'j~~^-\  is  a  factor  of/(r) ; 
if,  therefore,  these  two  expressions  have  no  simple  factor  in  common,  their 
product  will  be  a  biquadratic  factor  of /'(.')• 

(x'-K+^+O'r-^)-'. 
which  car  always  be  resolved  into  two  real  quadratic  factors.     (See  solution 
of  liiquad  atics.)     If  they  have  a  factor  in  common,  since  they  may  be  written 


QUADRATIC  FACTORS  OF  EQUATIONS.  435 


x>—>x+y-  V-l(dr-<5'),  x—  yx+y+  yf —l(&x-&% 
it  can  only  be  of  the  form  r — £ ;  and  the  factors  themselves  become 

[x—K+\<f^i)(x—e),  (x— k  —  ?.  /3T)(x— e)  ; 
and,  therefore,  the  proposed  equation  admits  the  real  quadratic  factor 

Hence  an  equation  whose  degree  =2r+V*  will  have  a  real  quadratic  factor, 
provided  an  equation  whose  degree  =2rfj.  has  one  ;  but  we  have  proved  this 
to  be  the  case  when  r=l  ;  therefore  it  is  universally  true  that  every  equa- 
tion of  an  even  degree  has  at  least  one  real  quadratic  factor.  If  now  this  fac- 
tor be  expelled,  the  depressed  equation  will  have  its  coefficients  real  and  its 
degree  even,  and  will,  therefore,  as  before,  have  one  real  quadratic  factor. 
Hence  the  first  member  of  every  equation  of  an  even  degree  may  be  resolved 
into  real  quadratic  factors. 

366.  Hence  if  we  divide  the  first  member  of  any  equation 
xn+p1in-1+^;rn-=+  .  .  .  +pa  =  0 
by  x2-|-a.r-f-6,  admitting  no  terms  into  the  quotient  that  have  x  in  the  de- 
nominator, we  shall  at  last  obtain  a  remainder  of  the  form  A.r-f-B,  A  and  B 
being  rational  functions  of  a  and  b;  and  in  order  that  x--j-ax+6  may  be  a 
quadratic  factor  of  the  proposed  equation,  it  is  necessaiy  and  sufficient  that 
this  remainder  should  equal  zero  for  all  values  of  x,  which  requires  that  we 
separately  have  A=0,  B  =0.  The  different  pairs  of  values,  real  or  imaginary, 
of  a  and  b  which  satisfy  these  equations  will  give  all  the  quadratic  factors  of 
the  proposed;  and  as  the  number  of  these  factors  is  \n(n  —  1)  (Art.  244,  Cor. 
2),  the  final  equation  for  determining  one  of  the  quantities  a,  b,  obtained  by 
eliminating  the  other  between  the  two  preceding  equations,  will  be  of  the 
degree  ±n{n— 1),  which  exceeds  n,  if  n>3  ;  therefore,  the  determination  of 
the  quadratic  factors  of  an  equation  will  generally  present  greater  difficulties 
than  the  solution  of  the  equation. 

As  the  proposed  equation  has  necessarily  \n  or  \{n— 1)  real  quadratic  fac- 
tors, according  as  n  is  even  or  odd,  there  will  always  exist  the  same  number 
of  pairs  of  real  values  of  a  and  b,  satisfying  the  equations  A=0,  B  =  0  ;  and 
if  any  of  these  pairs  of  real  values  be  commensurable,  they  may  be  easily 
found ;  and  the  commensurable  quadratic  factors  being  known,  the  equation 
may  be  depressed. 

EXAMPLES. 

(1)  To  resolve  a^— 6i2-L.7i£— 3=0  into  its  factors.  Dividing  by  i8-f  ax+b, 
we  find  a  remainder, 

{n  +  2ab+6a  — a3)x— {a-b— o2  —  66+3) ; 
therefore,  to  determine  a  and  b,  we  have 

n  +  2ab  +  Ga— a3=0, 
a?b  —  b2  —  6b  +  3  =  0. 
Solving  the  former  with  respect  to  b,  and  substituting  in  the  latter,  we  find 

(a3— 4)3= n3  —  64,  or  a=\'4-L-  fyn2  — 64 ;   from  whence  6,  and  the  other 
quadratic  factor, 

Is — ax+a"  —  6—6, 
may  be  determined. 


436  ALGEBRA. 

(2)  The  resolution  of  r4-r-/>r,-|-7r':+r-r4"5  into  its  two  quadratic  factors 
-3-r-m-r+">  x"-\-m'x-\-n,  may  be  effected  by  the  following  formulae  : 


in 


=?(?+  Vz),m'  =  \(2>-  y/z), 


r  —  am  -\-  pin2  —  m3  r  —  qm '  +  /'<'-'  —  m  '* 

n  = —^ ,  n'  = ■-      ,    . , 

p — 2m  j1  — 

wnere  z  is  a  root  of  the  equation, 

z3_(3p«_  8?)z9-j-(3p*— 16p8j-fl6os-T-16pr— 64s)z— (8r— 4pj+j>»)8~0, 

which  has  necessarily  a  real  root. 

elimination  BY  SYMMETRIC  FUNCTIONS. 

367.  Symmetric  functions  furnish  a  method  of  elimination  which  has  the 
advantage  of  making  known  the  degree  of  the  final  equation. 

Let  the  two  equations  be 

xm_|_pxm-l_J_Qa.m-2_^Rxm-3#.._0 (1) 

i»4.P'f-'^QV-2^.R'i»-»...  =  0 (2) 

in  which  P,  Q...,  P\  Q'. . .  are  functions  of  y.  If  we  could  resolve  (1)  witn 
respect  to  x,  we  would  derive  from  it  m  values,  a,  b,  c...,  of  x,  which  would 
be  functions  of  y ;  and,  by  substituting  these  values  of  x  in  equation  (2),  we 
would  have,  for  determining  the  values  of  y,  in  equations  free  from  x,  viz., 

an+P'an-1  +  Q'an--+R'an-3...=0  ^ 

6n4-P'60-1-fQ'6B-s-fR/6,t-3...=o[    ....  (3) 

C"_}-P'cn-l  +  Q'cn--+R'cn-;5...=:0) 
&c.  ckc. 

But,  in  general,  the  resolution  of  equation  (1)  is  impossible,  and  the  prob- 
lem is  to  obtain  a  final  equation  which  embraces  all  the  values  of  y  without 
distinction. 

We  shall  have  an  equation  which  will  fulfill  this  condition  by  multiplying 
together  the  m  equations  (3),  for  the  resulting  oquation  will  be  satisfied  by 
each  value  of  y  derived  from  any  one  of  them,  and  it  can  not  be  satisfied  in 
any  other  way.  But  the  factors  of  this  resultant  can  only  change  places, 
whatever  permutations  we  may  make  between  the  quantities  a,  b,  c  .  .  .  ;  the 
product,  then,  will  only  contain  entire  and  rational  symmetric  functions  of 
these  quantities  ;  hence  we  shall  bo  able  to  express  these  factors  by  means 
of  the  coefficients  of  equation  (1),  and  in  this  way  wo  shall  have  the  final  equa- 
tion in  y. 

This  method  of  elimination  leads,  in  general,  to  \  calculations, 

but  it  has  the  advantage  of  giving  a  final  equation  containing  all  the  roots  that 
it  ought  to  embrace,  without  any  complication  of  foreign  roots. 

368.  This  method  has  also  the  advantage  of  leading  to  a  general  the. 
with  respect  to  the  degree  of  the  final  equati         [n  the  ]  re  rti(  le  the 
first  equation  is  of  the  d<               the  second  of  the               .  and  P,  Q...,  I", 
Q'...  are  any  functions  whatsoever  of  y;  but,  for  the  theorem  in  question, 

■  functions  must  evidently  be  polynomes,  such  that  the  sum  of  tl 
ponents  of  .r  and  y  Bhall  be,  at  most,  equal  to  m  in  each  term  of  equation  \l), 
hiuI,  at  most,  equal  to  n  in  each  term  of  equation  (2).     We  have,  then,  to  de- 
termine to  what  degree  y  can  be  raised  in  the  symmetric  functions  which 
compose  the  product  of  equations  (3). 

Each  term  of  this  product  is  the  product  of  m  t  m  respectively  from 


ELIMINATION  BY  SYMMETRIC  FUNCTION'S.  437 

the  m  equations  (3) ;  hence,  designating  these  terms  by  Yaa,  Y'Zr,  Y"cy,  the 
term  of  the  product  will  be  YY' Y". . . aabPcr . . .  But  f  lie  product  of  these  m 
equations  being  symmetric  with  respect  to  the  quantities  a,  l>,  c...,  all  tho 
terms  should  have  tho  same  form  with  the  one  that  we  have  given  above  ; 
consequently,  we  know  that  the  product  embraces  all  the  terms  represent- 
ed by 

YY'Y"...X.S(a"^c>/...) (4) 

We  have  now  to  determine  the  degree  of  y  in  this  expression.  Observing 
that  the  degree  of?/  in  Y  is,  at  most,  equal  to  n  —  a,  in  Y'  to  n — ,',  in  V"  to 
n — y,  &c,  we  shall  readily  see  that  in  YY'Y". ..  its  degree  will  be,  al  most, 
equal  to  mn — a — j3 — y On  the  other  hand,  if  we  refer  back  to  the  rela- 
tions (Art.  35G)  from  which  the  sums  Su  S.,  S3,  tVc,  are  derived,  we  shall 
see  that,  P  being,  at  most,  of  the  first  degree  in  y,  Q  of  the  second,  R  of  the 
third,  and  so  on,  tho  degree  of  y  in  these  sums  can  not  surpass  the  subscript 
number  of  S  ;  and,  in  like  manner,  if  wo  refer  (Art.  359)  to  tho  formulas 
which  express  double,  triple,  &c,  functions,  we  shall  perceive  that  in 
S(«"i' <■'.. .)  the  degree  of?/  can  not  surpass  a-f-/3-(- }- . .  Hence  in  expres- 
sion (4)  the  degree  of  y  will  be,  at  most,  equal  to  mn. 

The  same  remark  will  apply  to  all  tho  symmetric  functions  whoso  sum 
composes  the  product  of  the  m  equations  (3)  ;  therefore,  lastly,  the  final  equa- 
tion can  not  be  of  a  degree  superior  to  mn. 

The  demonstration  seems  to  require  that  equation (1) contain  m.  Bet  wc 
can  suppose  that  at  first  .rm  had  a  coefficient,  A,  independent  of  y,  and  that  we 
have  divided  the  whole  equation  by  A.  The  final  equation  ought  to  subsist, 
whatever  may  be  the  value  of  A  ;  we  can  make  A=0,  and  it  is  evident  that 
this  supposition  will  not  raise  the  degree  of  the  final  equation.  Finally,  the 
theorem  is  to  be  thus  understood  :  that  tho  elimination  between  two  general 
equations,  the  one  of  the  degree  m,  the  other  of  tho  degree  n,  ought  to  give  a 
final  equation  of  the  degree  mn  ;  but  that,  in  particular  cases,  the  degree  of 
the  final  equation  can  be  loss  than  mn. 

EXAMPLES. 

Tho  two  eqiiations,  x — ym  =  0,  x"-\-ayn-\-by-\-c  =  0,  although  veiy  simple, 
will  give  a  final  equation  fully  of  the  degree  mn ;  for,  by  substituting  in  the 
second  the  value  of  x  derived  from  the  first,  it  becomes  ymn-\-ayn-{-by-\-c—(). 

On  the  other  hand,  in  eliminating  x  between  the  equations  xn — i/m=0, 
x"-\-ay"-}-by-\-c=0,  wo  obtain  a  final  equation  of  a  degree  less  than  mn,  viz., 
y™+ay"  +  by+c=0. 

369.  For  extending  the  theorem  to  any  number  whatsoever  of  equations, 
we  have  the  general  theorem  given  by  Bezout,  viz.,  that  If,  between  equations 
equal  in  number  to  that  of  the  unknowns,  ice  eliminate  all  the  unknowns,  except 
one,  the  degree  of  the  final  equation  will  be,  at  most,  equal  to  the  product  of  the 
degrees  of  these  equations. 

Before  Bezout,  the  theorem  had  been  known  for  the  case  of  two  equations  : 
and  Cramer,  in  the  appendix  to  his  Introduction  to  the  Analysis  of  Right 
Lines,  has  given  a  very  simple  demonstration,  which,  in  reality,  does  not  differ 
from  that  which  we  have  stated.  It  has  been  a  desideratum  that  tho  same 
demonstration  should  be  capable  of  being  applied  to  all  other  cases;  this  has 
been  accomplished  by  Poisson,  in  a  memoir  which  appeared  in  the  i  leventh 
volume  of  the  Journal  dc  VtlcoU   Polytt   '    '.que. 


138  ALUEB11A. 

METHOD  OF  TSOHIENHAUSEN  FOR  SOLVING  EQUATIONS. 

370.  As  another  application  of  the  theory  of  elimination,  we  shall  briefly 
illustrate  the  principle  upon  which  Tschirnhausen  proposed  to  accomplish  th« 
general  solution  of  equations,  but  which,  as  observed  at  Art.  277,  was 
found  to  be  of  but  very  limited  application,  not  extending  beyond  equations  ol 
the  fourth  degree  ;  and,  even  within  this  extent,  too  laborious  for  general 
The  principle  consists  in  connecting  with  the  proposed  an  auxiliary  equation 
of  inferior  degree  with  undetermined  coefficients,  and  of  as  simple  a  form  as 
possible  consistently  with  the  office  it  is  to  perform,  but  involving,  besides  the 
unknown  quantity  x,  a  second  unknown  y.  The  unknown,  common  to  both 
equations,  is  then  eliminated  according  to  the  method  at  Art.  315,  and  a  Final 
equation  in  y  thus  obtained,  of  which  the  coefficients  are  functions  of  the  un- 
determined coefficients  in  the  auxiliary  equation.  The  arbitrary  quantities, 
thus  entering  the  coefficients  of  the  final  equation  in  y,  are  then  determined 
80  as  to  cause  certain  of  these  coefficients  to  vanish;  by  which  means  the 
equation  is  ultimately  reduced  to  a  prescribed  form,  supposed  to  be  solvable  by 
known  methods. 

371.  As  an  example,  let  it  be  required  to  reduce  the  cubic  equation 

xri^-ax'2-{-bx-\-c=0     (1) 

to  the  binomial  form 

y*+k=0. 
Assume  an  auxiliary  equation 

x2+a'x+b'  +  y  =  0 (2) 

and  eliminate  x  from  (1)  and  (2)  in  the  usual  way.     The  remainder  arising 
from  dividing  the  first  member  of  (1)  by  the  first  member  of  (2)  is 

(a'2— aa'+b  —  b'  — y)x+(a'— a){b'+y)  +  c, 
which,  equated  to  zero,  gives 

(a-a')(b'+y)-c  . 
x—a'i—aa'+b—b'—y, 

and  this  value  of  x,  substituted  in  the  proposed  equation,  transforms  it,  after 
reduction,  into  the  form 

y*+hf-+iy  +  k=0 (3) 

where 

h—  3b'— aa'+a'2— 2b 
i=3b'2—2b'(aa'  —  a2-\-2b)-\-a'-b 
+(3c— ab)a'+b9— 2ae 
k  =  bri—ab'ia'  +  bb'a"i—ca'3+{a-—1b)b'*-)r 
(3c_a6)a'6,+aca'2+(ta— 2ac)b'  —  bra'  +  c*. 
Hence,  in  order  to  reduce  (3)  to  the  prescribed  form,  wo  must  determine 
the  arbitrary  quantities  a',  b'  conformably  to  the  conditions  /i  =  0,  i=0  ;  that 
is,  these  quantities  must  satisfy  tin-  equations 

3B'_  aa'+cP— 26=0 

3&'9— 2o'(oa'— a9+2&)4-a"o+ 

(3c— a6)a'+o9— 2ac=0, 

of  which  tho  first  is  of  the  firsl  degree  with  respect  to  a'  and  /-',  and  the  otuei 

of  tlu>  Becond  degree,  bo  thai  their  values  may  be  determined  by  a  quadratic 

aquation.     And  these  values,  or.  rather,  the  expression  for  them  in  terms  of 


METHOD  OF  LAGRANGE.  439 

the  given  coefficients,  being  substituted  in  the  preceding  expression  for  fc,  reo 
der  that  symbol  known  ;  and  thus  tho  required  form 

y3+k=0 
is  obtained. 

372.  In  a  similar  manner  may  the  general  equation  of  the  fourth  degree 

x* + ax*  +  bx* + ex + d = 0 
be  transformed  into  one  of  the  form 

y*+hy2+k=0, 
which  is  virtually  a  quadratic,  by  eliminating  x  from  the  pair  of  equations 

a:4  -f-  ^.r3  +  bx"  -\-  cx-\-  d = 0, 

x*+a'x+b'  +y=0, 
which  elimination  will  conduct  to  a  final  equation  in  y  of  the  form 

from  which  the  second  and  fourth  terms  will  vanish  by  the  equations  of  con- 
dition 

the  first  of  which  will  be  of  the  first  degree  as  regards  the  arbitrary  quantities 
a',  b',  and  the  second  of  the  third  ;  both  quantities  are,  therefore,  determina 
ble  by  means  of  an  equation  of  the  third  degree,  and  thence  the  quantities 
h,  Jc,  which  are  known  functions  of  them. 

All  this  is  very  laborious,  but  it  really  does  effect  the  object  proposed  thus 
far ;  that  is,  it  reduces  the  solution  of  equations  of  the  third  and  fourth  do 
grees  to  those  of  inferior  degrees  ;  but  beyond  this  point  the  method  fails,  as 
the  conditional  equations  resolve  themselves  ultimately  into  a  final  equation 
that  exceeds  in  degree  that  which  they  are  intended  to  simplify. 

On  this  subject  we  may  add  that  Mr.  Jerrard  has  greatly  extended  the  prin- 
ciple of  Tschirnhausen,  and  has  succeeded  in  reducing  the  general  equation 
of  the  fifth  degree 

z5+  A4r* + A^ + A,.r2  -f  Ax + N =0 
to  the  remarkably  simple  forms 

x5^-axi+b=0 

.r5 -far3 +6=0 

x5^-ax'i-\-b=0 

x5-\-ax  +6=0; 

so  that  tho  solution  of  the  general  equation  of  the  fifth  degree  might  be  con- 
sidered as  accomplished  if  either  of  the  above  forms  could  be  solved  in  general 
terms. 

For  a  very  masterly  analysis  of  Mr.  Jerrard's  researches,  the  reader  is  re- 
ferred to  the  paper  of  Sir  W.  R.  Hamilton  in  the  Report  of  the  sixth  meet- 
ing of  the  British  Association. 

METHOD  OP  LAGRANGE  FOR  SOLVING  EQUATIONS. 

373.  A  remarkable  application  of  the  theory  of  symmetrical  functions  is  that 
made  by  Lagrange  to  the  general  solution  of  equations  ;  by  that  means  he 
solves  the  general  equations  of  the  first  four  degrees  by  a  uniform  process, 
and  one  which  includes  all  others  that  have  been  proposed  for  that  purpose, 
the  common  relation  of  which  to  one  another  is  thus  made  apparent. 

It  consists  in  employing  an  auxiliary  equation,  called  a  reducing  equation, 
whose  root  is  of  the  form 


440  ALGEBRA. 

denoting  by  x_,  x:,  .  .  .r„  the  n  roots  of  the  proposed  equation,  and  by  a  one  of 
the  n']'  roots  of  unity;  and  the  principle  on  which  it  is  based  is  as  follows- 
Let  y  be  the  unknown  quantity  in  the  reducing  equation,  and  let 

y  =  alxlJra,r:-\ +  n„xa, 

d,  cia,  ...do  denoting  certain  constant  quantities;  then,  if  n  —  1  values  of  y, 
and  suitable  values  of  the  constants  o„  a,,  .  .  .  a,„  can  be  found,  so  that  we  may 
have  n  —  1  simple  equations,  these,  together  with  the  equation 

—lh  =  i-i+r;+  •••  +Xm 
will  enable  us  to  determine  the  n  roots. 

Now,  supposing  the  constants  in  the  value  of  y  to  preserve  an  invariable 
order,  aM  a2,  &c,  since  the  number  of  ways  in  which  the  n  roots  may  be  com- 
bined with  them  to  form  the  expression  aj£j-f-a>rs-f-,  cVc,  is  the  Bame  as  the 
number  of  permutations  of  n  things  taken  all  together;  therefore,  the  expres- 
sion for  y  \\\]1  have  71(71  —  1) . .  .  3.2.1  values,  and  the  equation  for  determining 
y  will  rise  to  the  same  number  of  dimensions,  or  will  be  of  a  degree  higher 
than  that  of  the  proposed  equation  ;  hence  the  method  will  be  of  no  use,  un- 
less such  values  can  bo  assumed  for  the  constants  a,,  a:, .  .  .  an  as  shall  make 
the  solution  of  the  equation  in  y  depend  upon  that  of  an  equation,  at  most,  of 
n  —  1  dimensions.  Now  this  may  be  done  (at  least  when  n  does  not  exceed 
4)  by  taking  the  nlh  roots  of  unity  a0,  a,  a8,  a\  .  .  .  an_1  for  a,,  a*. .  . .  aQ,  so  that 

y=za°Xi-{-ax.--{-  . . .  -|- ar-'xr -}- ar.rH  1 -f- j-aa-'.r0. 

For,  in  the  first  place,  with  this  assumption,  the  reducing  equation  will 
contain  only  powers  of  y  which  are  multiples  of  n ;  for,  since  an=l, 

an-ry  =  a°-*x1  +  a»-<+Kv.z+  . . .  +xr+i+axt+i+ j-tf"-*-^, 

or  aa~Ty  =  an.rr+1  +  a.rr+;  + \-  a»-Kcr, 

wliich  is  the  same  result  as  if  we  had  interchanged  xx  and  xr+i,  x2  and  x^, 
&c,  so  that  if  y  be  a  root  of  the  reducing  equation,  an~ry  is  also  a  root ;  there- 
fore, the  reducing  equation,  since  it  remains  unaltered  when  an-'y  is  written 
for  y,  contains  only  powers  of  y  which  are  multiples  of  n  ;  if,  therefore,  we 
make  yn=z,  wo  shall  have  a  reducing  equation  in  r  of  only  1.2.3  . . .  (n  —  1 
dimensions,  whose  roots  will  be  the  different  values  of  z  which  result  from 
the  permutations  of  the  n  —  1  roots  .?•..,  .r,,  . .  .  rn  among  themselves.  We  shall 
now  have,  expanding  and  reducing, 

z=yn  =  u0-\-ula-{-u2a-+  .  .  .  +tt0_iOn-1, 

in  which  w0.  «i,  v,,  .  .  .  m„_,  are  determinate  functions  of  the  roots,  which  will 
be  invariable  for  the  simultaneous  ehau  into  ..-r+1,  rs  into  xr+:,  cVrc, 

since  z  =  (ary)n  ;  and  when  their  values  are  known  in  terms  of  the  coefficient! 
of  the  proposed  equation,  Ave  shall  immediately  know  the  values  of  the  roots 
For  let  z0,  z\,  z,,  .  .  .  zn_,  be  the  different  values  of  :,  when  1,  c,  /?,  y,  . . .  A 
the  roots  of  i/n — 1=0,  aro  substituted  for  a  ;  then,  since  y=  V :,  we  have 

Xi+X%+   ■  .  .  +•'•„=  V£o 
*l  +  «•»••:+    •  •  •   +a"    '•'•„=  V-l 


therefore,  adding,  an  1  taking  account  of  the  properties  of  the  suiur  of  the 
powers  of  I.  a,  8,  y,  Sic.,  (Art.  .",.'>7,  [2]),  we  gel 


METHOD  OF  LAGRANGE.  4-H 

nXl=  V~o+  V?i+  • ,.  •  +  V~i- 

Again,  miltiplying  the  above  system  of  equations  respectively  by  1,  a"-1 
3"-',  .  .  .  A"-i,  we  get 


nx2=  Vz0+an-1^z1+/3"-1V-H +^_1Vzn-i, 

and  so  on  for  the  rest.     Hence,  since   — pi  =  Vzot  and  .-.  ( — j>1)n=z0r=M# 

+  MH \-ita— n  the  problem  is  reduced  to  finding  the  values  of  «,,  w;, . . .  u„_i. 

374.  When  n  is  a  composite  number,  the  above  general  method  admits  of 
simplifications.  For  let  n  have  a  divisor  m,  so  that  n=mp,  and  let  a  be  a  rool 
of  7/'"  — 1  =  0;  then,  since  am=l,  am+'=rt,  a!"+'  =  «2,  &c,  fl2m  =  l,  a^-Hssa 
-Sec,  we  have 

y=xi-\-avi-{-a"x.,-{-  .  .  .  +an^1x„ 

=X1+aXs+a*X3+ ham_1-Xmi 

where  Xr=.rr-r-.rm+r-f-.r2m+r-r-  .  .  .  +-r'i-i«+"  an(i  consists  of ^?  roots ; 

.-.  :=7/n  =  2i0+w1a-|-W2a24"  •  •  •  +Mm-i°m-1» 
where  w0,  Wi,  &c,  are  known  functions  of  Xu  X2,  &c. ;  and  when  they  ari 
found  in  terms  of  the  coefficients  of  the  proposed  equation,  wo  shall  be  able  to 
determine  immediately  the  values  of  Xj,  X2,  &c,  as  before.  To  deduce  tho 
values  of  the  primitive  roots  xu  x%,  xs,  .  .  .  .r„,  we  must  regard  separiitely  those 
which  compose  each  of  the  quantities  Xj,  X2,  &c.,  as  the  roots  of  an  equa- 
tion of  p  dimensions.  Thus,  let  the  roots  whose  sum  is  X!  be  those  of  the 
equation 

xp— XiZP-^  +  La*-2— Mxp-3+  .  .  .  =0, 
where  L,  M,  &c,  are  unknown  ;  then  the  first  member  of  this  equation  is  n 
divisor  of  the  first  member  of  the  proposed,  since  all  its  roots  belong  to  tho 
hitter.  Hence,  effecting  the  division  and  equating  to  zero  the  coefficients  of 
rP_1,  a:P~2,  &c,  in  the  remainder,  we  shall  have  p  equations  in  Xi,  L,  M,  &c, 
of  which  the  first  p —  1  will  give  the  values  of  L,  M,  &c,  in  terms  of  X,  by 
linear  equations.  It  will  then  remain  to  solve  the  equation  so  formed  of  p 
dimensions.  Similarly,  substituting  the  value  of  X2  in  place  of  that  of  Xu  we 
shall  have  an  equation  giving  tho  next  group  of  roots  .r2,  .t^j,  &c. ;  and  so  on 

EXAMPLE  I. 

Xs — px~-\-qx — Tz=0. 
Let  the  roots  be  a,  b,  c,  and  let 

y=a-\-al^-a-c  ; 
.-.  zz=zy*=a?+b*-{-c*+6abc+ 3(a2i  +  o2c+c2a)a+3(a2c+&2a-f  c2Z>)a«, 
=ua-{-u,a-\-2tia'. 
But  «i,  Ui  are  roots  of  the  quadratic 

u-  —  {ni  +  !<;)"  +  w1w.2  =  0, 
and  ul-{-u.2=3^(aib)  =  3pq— 9r  (Arts.  357,  359), 
ulu,=0\abcS3+Z{aW)+3a°b°c'i\ 
=  973+  9(pi  —  6pq)r+  Sir2. 
Hence  «i,  u2  are  known, 

and  .••  itos^p3  —  ("i  +  W;),  is  known. 
Hence,  denoting  by  z:,  zs,  the  values  of  z  when  a  and  a2  are  respectivef) 
written  for  a,  we  have 


ALGEBRA. 

a-\-b-\-c—p 
a-\-ab-\-a"-c=yTL 
a-\-a"b-\-ac=  ^/z2; 
from  which  we  obtain  the  values  of  a,  b,  and  c,  viz., 

EXAMPLE  II. 

x*  — px?  -f-  qx* — rx  -j-  s = 0 . 
Since  4=2.2,  let  a  be  a  root  of  y-  — 1=0,  so  that  <z2=l  ; 
then  3/=r1  +  ar2+T34-aj:4=X1-r-aX2, 

if  Xi=:.Ti-^-ar3,  Xo=.T;-}-:r.j ; 

•.  z=y2=u0-\-au1 

where  «0=X14-X^,  w1=2XiX!,  and  u0+Ui=z0—p'>' 

Hence  ul=2(xl-\-x3)(x2-\-Xi),  by  interchanging  the  roots  among  themselves, 
Will  admit  the  two  other  values  2(.r1+x2)(.r3-f  ar4),  and  2(.r1+x<)(.r.;:-r-x3),  and 
will,  therefore,  be  a  root  of  an  equation  of  the  form 

«J— MttJ+Ntt,— Pr=0; 

the  coefficients  being  symmetrical  functions  of  xu  x2,  x3,  x4,  and,  consequently, 
assignable  in  terms  of  p,  q,  r,  s.  It  is  easily  seen  that  if  we  make  ul=2q — 2u, 
we  shall  have  an  equation  in  u  whose  roots  are 

X^+XiXi,  .r^  +  .Ts-r.,,  XiX4-L.X2X3  ; 

and  the  transformed  equation  is  (Art.  362) 

us—qu'i-\-{pr—As)u  —  {2r—iq)s—ri=0. 
Let  u'  be  a  root  of  this  equation,  then  u^=2q — 2w' ;  hence,  making 
a=— 1,  z,  =  m0  —  U\=P'2  —  2ui=])-  —  Aq-\-Au'  ; 
.-.  Xl  +  X,=]y2_Xi— X:=  <fz~x\_ 

Hence  xl,  x3  may  be  regarded  as  roots  of  a  quadratic  x2 —  X,x-j-L=0; 
dividing  the  proposed  by  this,  and  putting  the  first  term  of  the  remainder  equal 
to  zero,  we  find 

x;-j>x;-HX,-r 

L-  2X,-p  ; 

therefore,  x,,  x3  are  known;  and  x2,  x4  will  result  from  tlie  same  forinuhe 
by  interchanging  X,  and  X2,  or  by  changing  tlie  sign  of  the  radical  -y/-i- 

EXAMPLE  III. 

xn  —  1 


-  =  0,  n  being  n  prime  number. 

If  r  be  one  of  the  roots,  and  a  be  a  primitive  root  of  the  prime  number  n 
(that  is,  a  Dumber  whose  several  powers  from  1  to  n — 1,  when  divided  by  n, 
leave  different  remainders),  it  will  be  proved  hereafter  that  all  tlie  roots  of 
his  equation  may  be  represented  by 

r,r>.  i to"-11. 

L,>r  yssr+ora+aVH l.  on- «r»n-,l 


METHOD  OP  LAGRANGE.  413 

a  beiiig  a  root  of  the  equation  yn~l  — 1=0.  Therefore,  observing  that  a°-l=.  1 
and  r"=l, 

z=2/n_1  =  M0-J-nM,+a2ML,-|-  .  .  .  +a"~2Mu-2,  ....  (1) 
u0,  m,,  &c,  being  rational  and  integral  functions  of  r  which  do  not  change  hy 
the  substitution  of  r",  r«3,  raS,  &c,  in  the  place  of  r;  for  these  quantities,  re- 
garded as  functions  of  xx,  x„,  x3,  &c,  do  not  alter  hy  the  simultaneous  changes 
of  x,  into  x2,  .To  into  x3,  &c,  nor  by  the  simultaneous  changes  of  x,  into  x3, 
,r2  into  x4,  &c,  to  which  correspond  the  changes  of  r  into  r",  into  r>*2,  &c. 

Now  every  rational  and  integral  function  of  r,  in  which  rn  =  l  may  be  re- 
duced to  the  form 

A  +  Br-fCr2-f  Dr5-}- p-Nr"-1, 

the  coefficients  A,  B,  C,  . . '.  N  being  given  quantities  independent  of  r ;  or, 
since  in  this  case  the  powers  r,  r2,  r3,  . . .  rn_1  may  be  represented,  although 
in  a  different  order,  by  r,  r«,  r«2,  .  . .  r«n_2,  we  may  reduce  every  rational 
function  of  r  to  the  form 

A-fBr+Cra-f-Dr«24-  ...  +Nr«"-2. 
Therefore,  if  this  function  is  such   that  it  remains  unaltered  when  r  is 
changed  into  ra,  it  follows  that  the  new  form 

A+Bra-fCra2  -f  Dr^-J +  Nr, 

coincides  with  the  preceding; 

.-.  B  =  C,  C  =  D,  D=E,  &c,  N=B, 
and  therefoi*e  the  function  is  reduced  to  the  form 

A+B(r+r<-  +  r«':!+ \-r«n--),  or  A— B, 

since  the  sum  of  the  roots  —  —  1  ;  hence  each  of  the  quantities  w0,  ult  «.:, 
&c,  will  be  of  the  form  A — B,  and  its  value  will  be  found  by  the  actual  de- 
velopment of  z=?/n-1 ;  so  that  we  have  the  case  where  the  values  of  w0,  ut,  n:, 
&c,  are  known  immediately,  without  depending  upon  the  solution  of  any 
equation.  Hence,  if  we  denote  by  1,  a,  /3,  y,  &c,  the  n  —  1  roots  of  the  equa- 
tion xn~l  — 1=0,  and  by  z0,  zu  z2,  &c.,  the  value  of  z  answering  to  the  substi- 
tution of  these  roots  in  the  place  of  a  in  equation  (1),  we  shall  have,  as  in  the 
former  cases, 

"--^+-^+-^1;+ . . .  +n-y"zZi 
r= ^r~ 

an  expression  for  one  of  the  roots  of  the  equation  xn  — 1  =  0  ;  and  the  othei 
roots  are  r2,  r3,  &c. 

Thus,  the  solution  of  xn — 1=0  is  reduced  to  that  of  the  inferior  equation 
2/n_1  — 1  =  0,  of  which  1,  a,  /?,  y,  &c,  are  the  roots  ;  also,  since  n  —  1  is  a  com 
posite  number,  the  determination  of  «,  [i,  y,  &c.,  will  not  require  the  solution 
of  an  equation  of  a  higher  degree  than  the  greatest  prime  number  in  n  —  1 ; 
that  is,  the  solution  of  xn — 1=0  (n  prime)  may  be  made  to  depend  upon  the 
solution  of  equations  whose  degrees  do  not  exceed  the  greatest  prime  number, 
which  is  a  divisor  of  n  —  1. 

EXAMFLK   IV. 

x5— 1  =  0.       • 
The  least  prim  itive  root  of  5  is  2 ;  for  the  powers  of  2  from  1  to  4,  when 
divided  by  5,  leat  e  remainders  2,  4,  3,  1  ; 


144  ALGEBRA. 

• 
.•.  1/  =  7•+a/--  +  a3/-,  +  a:,r,; 

also  ^=1,  r''=l,  and  r  + r:-\-r*-{-r'  = — 1; 

.-.  z=y*=  —  l  +  4a+14a-  — 16a?. 

But  the  four  roots  of  y4—  1  =0  are 

1,  -1,  V"l,  -  V^l ;  

.-.  Zb=l,  ^  =  25,  z2=  -15  +  20  v  — 1, 
z3=  — 15  —  -20  V—  l; 

•••  .r=}{—  1+  V5  +  V  — 15  +  20-/—  1  +  V  — 15—  20V3lf. 

375.  For  the   proof  that,  in  the  general  equation  of  the  »"■  degree,  the 

formation  of  the  reducing  equation  will  require  the  .solution  of  au  equation  of 

1  .2.3...  n 

1.2.3...  (n — 2)  dimensions,  when  n  is  prime  ;  and  of  ■; ■ 

'  '  (to— l)m(1.2.3...j>)m 

dimensions,  when  n  is  a  composite  number,  and  =zmp,  where  m  is  prime  ; 

and  that,  consequently,  the  method  fails  when  n  exceeds    1.    the    reader  is 

referred  to  Lagrange's  Trade  de  la  resolution  dcs  equations  nunuriques,  note 

xiii.,  from  which  the  matter  of  this  section  is  taken. 

RESOLUTION  OF  THE  GENERAL  EQUATIONS  OF  THE  THIRD  AND 

FOURTH  DEGREES. 

RESOLUTION    OF    THE    EQUATION    OF    THE    THIRD    DEJREE. 

376.  I  shall  suppose  that  wo  have  made  the  second  term  of  the  equation  of 
the  third  degree  disappear,  and,  to  avoid  fractious,  I  will  write  this  equation 
under  the  form 

x3+3px+2q=0 (1) 

Among  the  different  modes  of  resolving  it,  the  most  simple  consists  in  form- 
ing a  priori  an  equation  of  the  third  degree,  without  a  second  term,  which  ad- 
mits of  one  known  root,  but  expressed  with  indeterminates,  and  to  make  use 
afterward  of  these  indeterminates  to  render  the  equation  identical  with  the 
proposed  equation  (1).      To  establish  this   identity,  it  will  be  net  y  to 

write  two  equalities,  and  for  this  reason  we  employ  two  indeterminates. 

Let  there  be  made  x=a-\-b :  the  cube  will  be  x3=a3-\-b3+3ab(a-{-b)  ; 
then,  replacing  a-\-b  by  ar,  and  transposing,  we  shall  have 

x3—3abx—a3—b3=0 (2) 

an  equation  which  admits  the  root  x=a-\-b,  and  which  it  is  necessary  to  ren- 
der identical  with  equation  (1).     Therefore  wo  place 

ab=—  p,  a3+b"=  —  2<y  .  .  .  .  (3) 

Tho  first  of  these  equalities  gives  asb3=—j>\  Tims  we  know  the  sum 
a3+b?;  and  tho  product  d'b\  Then  the  values  of  a9  and  b3  aro  roots  of  an 
equation  of  the  second  degree,  in  which  the  coefficient  of  tho  second  term  is 
equal  to  +27,  and  tho  last  term  equal  to  —j>3  (seo  Art.  101) ;  so  that  this 
equation  will  be,  calling  ;  the  unknown, 

+  '-V— q>'  =  0. 
This  is  called  tho  redact  J  equation. 

lis  two  roots  represent  the  values  rf«*  and  6»j   moreover,  we  can  take' 
or  of  them  indifferently  for  the  valuo  ofo8,  because  this  amounts  to  chang- 
1  h.  and  b  into  a,  in  tho  valuo  x=a  +  b.     I  will  take 


\ 

EQUATION  OF  THE  THIRD  DEGREE.  445 


Each  radical  of  t\ie  second  degree  here  has  but  one  value,  but  each  one  of 
the  third  degree  has  three.     If  we  could  satisfy  equation  (3)  without  making 
any  choice  between  these  values,  we  could  also,  by  the  same  values,  render 
equation  (1)  identical  with  equation  (-2) ;  and  since  a-{-b  is  a  root  of  the  sec 
ond,  the  first  ought  to  be  satisfied  by  taking 

*=V-?+  Vf+i*,+\l-q-  Vq*+i*  •  •  (4)  • 

which  is  the  formula  of  Cardan. 

But  an  important  remark  presents  itself:  it  is,  that  since  each  radical  of  the 
third  degree  has  three  values,  the  above  expression  must  have  nine,  while 
the  equation  (1)  ought  to  have  but  three  roots.  It  is  necessary  to  explain, 
then,  whence  comes  this  multiplicity  of  values,  and  to  discern  among  them 
which  ought  to  be  true  roots  of  the  equation  (1). 

For  this  purpose,  let  us  observe  that,  properly  speaking,  it  is  not  the  reso- 
lution of  equations  (3)  which  has  given  a  and  b,  but  rather  the  equations 
a?b3=— p\  as-\-b3=—2q  ...  (5) 

Now  if  we  designate  by  a  and  a?  the  two  imaginary  cubic  roots  of  unity, 
which,  as  we  know,  are  the  one  the  square  of  the  other,  it  will  be  readily 
seen  that  the  equation  «:!in=  — p*  may  result  indifferently,  from  raising  to  tho 
cube  these  following : 

ab= — p,  ab= — np,  ab  =  — n-p. 

Hence  it  follows  that  the  nine  values  contained  in  formula  (4)  ought  to  give 
the  roots  of  the  three  equations, 

a?+3px+2t/=0,  s?+ 3opx+2g=0,  x8+3a'*px+2q=0  ....  (6) 

We  can,  moreover,  consider  these  nine  values  as  the  roots  of  the  equation 
of  the  9°  degree,  which  would  be  obtained  by  multiplying  together  the  three 
equations  (6).  But  it  will  be  more  simple,  and  will  amount  to  the  same  thing, 
to  raise  to  the  cube  either  one  of  these  equations,  after  transposing  to  the 
second  member  the  term  which  contains  p.     In  this  manner  we  find  at  once 

(&+2q)3=  — 21  fa*. 

As  to  the  roots  which  belong  especially  to  each  of  the  three  equations,  what 
precede^  furnishes  the  means  of  distinguishing  them ;  because,  according  as 
the  coefficient  of  x  shall  be  3p,  Zap,  or  Za-p,  it  is  clear  that  we  ought  to  add 
only  the  values  of  a  and  b,  for  which  we  have  abz= — p,  or  ab  =  —  ap, 
or  ab  =  —a'2p. 

By  this  rule  it  will  be  easy  to  form  the  roots  of  the  proposed  equation 
r1-\-3px-\-2q=0,  the  only  one  with  which  we  have  to  do.  Designate  by  A 
one  of  the  values  of  the  first  cubic  radical,  and  by  B  ono  of  the  values  of  the 
second  ;  tho  values  of  a  and  b  will  be 

g  =  A,  aA,  cfiA;  6=B,  aB,  a:B. 

Moreover,  suppose,  for  {hurts  admissible,  that  A  and  B  represent  the  values, 
the  product  of  which  is  — p.  From  what  has  just  been  said  we  ought  to  add 
only  the  values,  the  product  of  which  is  AB ;  then,  recollecting  that  a,=l. 
we  must  take 


446  ALGEBRA. 

x=A  +  B,  i=oA+a?B,  x=a2A+aB; 
and,  besides,  we  know  (3U3)  that  we  have 

a  = ,  a>  = 

II*  we  replace  A  and  B  by  the  two  cubic  radicals,  and  a  and  a*  by  thej 
va.  ues,  we  shall  have 


=V-<7+  Vf+P3+\!-q- 


Vt+p3, 


g=-i+v-3^_g+  ^+^+ -1  -  v  -y  _g_  v^+/a 


-l-V- 


-y[--q+  v^+F'+~1+/~y-7-  Vg*+f»- 


These  are  the  roots  of  the  proposed  equation,  but  we  must  take  care  to  at- 
tach to  the  two  cubic  radicals  the  same  restricted  sense  as  to  A  and  B,  with- 
out which  we  should  find  false  roots. 

377.  To  discuss  these  valves,  it  will  be  more  convenient  to  leave  A  and  B 

substituted  for  the  cubic  radicals,  and  to  isolate  the  one  which  is  multiplied  by 

V — 3.     By  this  means  we  have 

x=A  +  B, 

A  +  B     A— B    _ 
ar=— f-  +— J-  V3, 

A+B     A— B    - 

sss — 2 — ~ T~V3- 

I  shall  suppose,  also,  as  is  done  ordinarily,  that  tho  coefficients  3p  and  Mcj 
represent  real  quantities.  Then  equation  (1),  being  of  an  uneven  degree,  has 
always  one  real  root,  and  it  is  admissible  to  suppose  that  A  and  B  are  the 
values  of  a  and  b,  which  give  this  root;  so  that  A  +  B  will  be  a  real  quantity. 
This  being  premised,  let  us  return  to  the  two  radicals 


!=V—  ?+  Vf+f,  b=\]  —  q—  vAyH 


a=y  —  </+  Vq2+p\  6=V  — <7—  vr+r'1- 

If  q"-{-p3^>0,  each  of  them  has  one  real  value  ;  then  we  can  suppose  A  and 
B  real.  Consequently,  A  +  B  and  A  — B  will  be  so  also;  then  the  first  root 
x=A+B  is  real,  and  the  other  two  are  imaginary. 

If  <72+j?3=0,  we  have  A  =  B,  and  then  the  three  roots  will  be  Cas2A, 
x= — A,  x= — A.  They  are  all  three  real,  and  the  last  two  are  equal  with 
one  another. 

Finally,  let  72+p3<[0,  which  requires  p  to  bo  negative.  Then  a  and  b 
have  no  longer  any  real  determination,  and,  consequently,  the  three  values  of 
X  are  found  complicated  with  imaginary  quantities.  However,  we  know  that 
one  of  them  must  bo  real,  and,  indeed,  it  is  evident  that  tho  cases  in  which 
the  three  roots  of  equation  (1)  are  real  and  unequal  can  only  be  found  on  tho 
hypothesis  in  question,  thai  7  +/''<j>,  as  may  be  seen  by  referring  to  the 
supposition  just  above  of  59+p3">0.  It  would  be  wrong,  then,  to  affirm  that 
the  values  of  x  are  imaginary.  I  will  prove,  in  fact,  that  neither  of  them  are 
so;  and  as  we  can  always  suppose  thai  A  and  B  are  determinations  such  that 
the  sum  A  +  B  represents  the  real  root,  the  existence  of  which  is  demon- 
strated, u:e  whole  is  reduced  to  showing  that  the  pari  '.(A  —  B)  v — 3,  which 


EQUATIONS  OF  THE  THIRD  DEGREE.  447 

is  found  in  the  other  two  values  of  x,  must  be  real.     By  the  rules  of  algebra 
alone  we  have  (A— B)(A2+AB  +  B2)=A3— B3 ;  then 

A3— B3  A3— B3 

A_B— A2+AB  +  Bi  —  (A+B)2^AB" 


But,  because  of  the  values  of  a3  and  of  i3,  we  have  A3 — B3=2  V(f-\-p3\  and, 

by  the  manner  in  which  A  and  B  have  been  chosen,  we  have  AB  =  — p ; 

2  v/ry24-»3 

then,  making  A+B=x',  there  results  A — B= — — ;  consequently 

x   -j-p 


A-B    —      j-.i{cf-+f) 
2      V— J-        x,i+p 

But  by  hypothesis  we  have  q"-^-pa<^0  ;  then  the  quantity  above  is  real ;  then 
the  three  values  of  x  are  also. 

It  is  thus  demonstrated  that,  upon  the  hypothesis  of  q2-\-]?K.b,  the  imag- 
inary quantities  which  affect  the  three  values  of  x  must  destroy  one  another. 
It  would  seem,  therefore,  that  analysis  ought  to  furnish  the  means  of  making 
them  disappear,  but  as  yet  it  has  not  been  found  capable  of  effecting  this  re- 
duction. For  this  reason,  the  case  under  examination  has  been  called  the  ir- 
reducible case.  Whenever  the  equation  falls  under  this  case,  the  general  ex- 
pressions of  the  roots  will  be  of  no  use  in  calculating  their  numerical  values. 
and  then  we  can  recur  to  the  methods  of  Arts.  290-297. 

EXAMPLES. 

(1)  r3— G.r— 9=0. 

9  7 

We  have  j9  =  — 2,  q= — -  .-.  ■\/q--\-p[i=-,  which  gives 


•W= 


16 


9+  Vqi+Pi=y-7T=2, 


B=\J-q-  Vq>+f=\ll=l. 


Thus  the  three  roots  are 
x=3, 


*=-!+  V3V-l  +  J(-l-V3V-l)  =  '(-3+  V3V-1), 

(2)  x3— 21x+20  =  0. 

Here  p=— 7,  9=10; 

.•.x=\j  — 10  +  9  V^+V— 10— 9-/^3. 
This  example  is  one  of  the  irreducible  case.  The  general  value  of  x  ap- 
pears in  an  imaginary  form,  and  yet  the  roots  are  real,  being  the  numbers  1, 
4,  and  — 5,  which,  by  substitution,  will  be  found  to  verify  the  given  equation. 
378.  The  solution  of  the  irreducible  case  may  be  obtained,  also,  by  the  help 
of  a  table  of  sines  and  cosines.  We  subjoin  the  method,  for  the  benefit  of  the 
student  acquainted  with  trigonometry. 

Solution  of  the  irreducible  case  by  trigonometry. 
;os  20=2  cos-  0—1 
cos  30=2  cos  20  cos  0 —  cos  6 


44ft  ALGEBRA. 

Substituting  the  first  expression  in  the  second, 

cos  3*9=4  cos3  0—3  cos  0. 
Whence 

3  ! 

cos3  0— -  cos  6— -  cos  30=0 (1) 

4  4  x  ' 

In  the  proposed  cubic  equation,  which  we  may  write  under  the  form 

xa+3px+2q  =  0 .- (2) 

x 
put  the  unknown  r  cos  0  for  x ;  or,  which  is  the  same  thing,  put  -  for  cos  6 

und  (1)  becomes 

3  1 

x3— -r-x— -r3  cos  30=0. 

4  4 

Comparing  this  with  (2),  we  have 

1 

-r3  cos  30=—  2q, 

and 


-r2=  —  3p  .-.  r=2  -y/  — p,  which  is  real,  p  being  negative  , 

2o  q 

.-.  cos  30=-  ' 


pr       j  _ps 

Consequently,  the  trigonometrical  solution  of  the  proposed  cubic  equation, 
that  is,  the  determination  of  0,  and  thence  of  r  cos  0,  depends  upon  Qie  triscc- 
lion  of  an  arc,  or  the  determination  of  cos  0  from  cos  30. 

The  mode  of  proceeding  by  aid  of  trigonometrical  tables  is  obvious  ;  we  are 

to  seek  in  the  table  of  cosines  for  the  angle  whose  cosine  is  q-J jj  this  will 

be  the  angle  30,  and,  consequently,  one  third  of  it  will  be  0 ;  and  the  cosine  of 
this,  multiplied  by  r,  or  2  -/  — p,  will  give  r  cos  0=x  for  one  of  the  real  roots 

of  equation  (2).     As  the  given  cosine,  J-v/'^-y  belongs  equally  to  three  arcs 

viz.,  30,  2n-\-3d,  and  2tt — 30,  by  taking  the  cosine  of  one  third  of  each  of  the 
latter  two,  wo  shall  have  the  values  of  the  remaining  roots.  Thus  all  the 
three  roots  will  be  expressed  as  follows : 

2  Vr^  cos  0,  2  y/— P  cos  -(2tt+30),  2  V  — p  cos  -(2~— 30). 

Or,  using  the  supplements  of  the  two  latter  arcs  instead  of  the  arcs  themselves, 
and  remembering  that  the  cosine  of  an  arc  is  equal  to  minus  the  cosine  of  its 
supplement,  we  have  somewhat  more  simply  (he  three  values  of  x  in  the  fol- 
lowing form  : 


2  V—  p  cos0,  —  2-v/—  p  cos  (G0°—  0),  —2-/—/'  co3(6O°  +  0). 
This  method,  with  a  single  exception,  applies  to  the  irreducible  case;  for. 
as  the  trigonometrical!  cosino  of  an  arc  is  always  less  than  unity,  except  when 
thai  arc  is  a  multiplo  of  180°,  we  must  have 


»vi 


,,.<»    .:q:<-p\ 

or  f+p*<0. 

When  30  is  a  multiple  of  180°,  two  roots  must  be  equal. 
The  reducible  case  may  also  employ  the  aid  of  trigonometry. 


EQUATIONS  OP  THE  THIRD  DEGREE.  449 

379.  If  in  the  expression 


we  put  cot  tpssA-J  ,  it  becomes  -JU—  cot  ^±  cosec  $)L 
Hence,  reducing,  the  real  root  of  ri-\-qx-{-r=0  is 

which,  by  putting  tan  -=  tan3  0,  may  be  further  transformed  into 


-4 


cot  26. 


q3     r- 
Similarly,  the  real  root  of  x3— qx-\-r=0,  -^z  <j,  becomes  (by  putting  cosec 

3 


7-/3V2-  ^ 

*=oU  '  tan  2=  tan3  0), 


\q/ 

la 

cosec  20. 


-4 


380.  The  following  method  of  arriving  at  a  now  and  valuable  formula  for  the 
solution  of  cubic  equations  will  be  found  an  excellent  exorcise  for  the  student  :* 
Let  the  given  equation  be 

&+px+q  =  0 (1) 

Placing 

x=m+y (2) 

we  obtain 

y3-\-3imf-\-(3m?-{-p)y-\-m3-{-pm+q=0 (3) 

Taking 

y=\ (4) 

we  obtain 

\l)  +3m(\y  +  (2m*+py:+7rv>+pm+q=0  ; 
which  gives 


3m2+p  3m  1 

1  m 
Placing 


z3+   , ,        ,    224—T-i r-z+-r~, r-=°  •  •  • (5) 

1  m3-\-pm-\-q     '  ms-{-pm-\-q    '  ?Ji3-\-pm-\-q  v  ' 


z-w-       3m2+i?         •  (6) 

3(?n3 -\-pm-\-q)  '  '  '  \  i 

we  find 

3p?n*+9qm—  p*        —  27qm3-{-18p"ni2-\-27pqm+27q'2-\-2p'i_ 

W*+  3{m3+pm+qyW+  27  (ma+pm+q)3  =°      *  '  '7) 

*  It  is  the  production  of  an  old  pupil  of  the  author's,  Mr.  James  S.  Woolley,  whom  ill 
health,  and  other  discouraging  circumstances,  have  not  prevented  from  making-  some  im- 
portant discoveries  in  alycbra,  which  it  would  be  premature  at  present  to  publish  to  the 
world. 

F  F 


150  ALGEBRA. 

The  value  of  m,  which  renders  the  coefficient  of  w  zero,  may  be  foi  nd  thus 

3pm8+9«/m— p»r=0. 
Then 


3<7  ,  3            1          «• 
m  =  -^,X^;+I 

The  value  of  w  in  (7),  substituting  the  value  of  m,  found  in  (8),  is  ezpree 
in  the  foUowing  four  equations,  (9),  (9,  a),  (9,  b),  (9,  c),  the  last  three  being 
obtained  by  decomposing  (9)  into  factors. 


1C: 


xv. 


w. 


(8ig+i2)(-K/J^f) 
(«£-»)(-W±*4) 


(9) 


(9,  a) 


(9,6) 


tV-|±7^3+tX\/1^+81| 

?i?—_ — £— 


(9,e) 


Substituting  in  (G)  the  values  of  ?ra  and  w,  found  in  (8)  and  (9,  c),  we  shall 
have 

(81I+12)(-IS&?) 

Substituting  in  (4)  the  values  «f  z,  given  in  (10),  and  decomposing  one  more 
nf  its  terms  into  factors,  we  shall  have 

■■Kjs^g_ 

Hence 


.y?&*f' '  /,'-',)  v^rn  >  vfe"+")! 


+ 


(continuing  the  numerator)  (8l!^18)  (_ f ±\/a7+ ?) (ia) 


EQUATIONS  OF  THE  THIRD  DEGREE.  451 

But  the  first  term  in  the  numerator  of  (12)  may  be  transformed  thus : 

And  the  last  term  in  the  numerator  of  equation  (12)  is 
Therefore  the  sum  of  the  first  and  last  terms  of  the  numerator  of  (12)  is 
Therefore, 


p'\27J-     '    4\      2~V27J      '    4/    '    V      *    '       p* 

54 

7 


54  n      g» 

Dividing  both  numerator  and  denominator  by  -p^2\/o7^?3~^T'  we        e 


I=(_l±%/^f)+l(_l±^^l)i 

The  numerator  of  this  value  of  x  is  equal  to 


The  denominator  is  equal  to 

Dividing  numerator  and  denominator  by  the  common  factor,  we  have 


x=- 


(-^V^ 


4V 


27^+4 


This  formula  may  be  reduced  to  that  of  Cardan  by  dividing  the  numerator 
by  the  denominator,  and  observing  that 

we  thus  obtain 

— HW^+9*+ (Hi  W^+iT- 

But  the  first  form  is  preferable,  as  it  gives  only  the  three  values  which  satisly 


452  ALGEBRA. 

equation  (1),  whereas  Cardan's  formula  gives  nine  values,  six  of  w  aich  have 
to  be  rejected. 

A  partial  division  gives 


1 

r\ i. 

3\3 


•={-1+     P-4-t) 

K-Wfe+9 


which  is  an  advantageous  form,  inasmuch  as  but  one  third  root  has  to  bo  ex- 
tracted, both  radicals  having  the  same  form. 

A  shorter  solution  of  the  above  might  bo  given,  but  we  have  already  extend- 
ed our  article  on  cubics  sufficiently  far. 

IRRATIONAL    EXPRESSIONS    ANALOGOUS    TO   THOSE    OBTAINED    IN    THE    RESO- 
LUTION  OF  EQUATIONS    OF   THE   THIRD    DEGREE. 


is  V  A±  y/B  ; 


381.  One  of  these  expressions  is"yAi  -\/B  ;  but  it  frequently  happens 
that  A  and  B  are  rational  numbers,  and  then  it  may  be  possible  to  reduce 
these  radicals  to  simpler  expressions,  in  which  there  are  no  longer  radicals 
over  radicals.  This  problem  has  already  been  resolved  for  radicals  of  the 
second  degree,  and  it  is  now  proposed  to  resolve  it  with  reference  to  radicals 
of  the  higher  degrees. 


VA+VB. 


I  shall",  commence  with  the  cubic  radical  y  A+  VB.     We  can  not  suppose 
for  this  root  a  quantity  of  the  form  V a-\-  y/b,  for  we  have 

(  V&+  "/*)'=«  Va+3«y6-f  3&V"a  +  6  Vb 
=  (a+3b)  V"a  +  (3a  + J)  -Jb, 

a  result  which  contains  the  radicals  V 'a  and  -Jb.  But  the  preceding  calcula- 
tion shows  that  we  should  have  a  result  of  the  form  A-j-  \/B,  by  raising  to 
the  third  power  the  expression  a-\-  y/b  and  (a-j-  V  b)  yc.  I  will  choose  this 
last  expression  as  the  more  general ;  we  shall  then  have 


-4, 


A+VB  =  (a+V6)Vc (1) 

Raising  both  members  to  the  thud  power,  it  becomes  A-|-  -v/B  =  c(a3+3a6) 
+c(3a3-f  &)-/&>*  equating  tho  rational  parts  together,  and  the  irrational  parts 
by  themselves. 

A=c{a*+3ab) (2) 

VB=c{3a°-+ b)  yfb (3) 

The  problem,  then,  is,  to  find  for  a,  b,  c  rational  values  which  satisfy  these 
two  equations.  But  squaring  these  equations,  and  then  subtracting  the  one 
from  the  other,  we  have 

A2— B=c2(a6— 3a<i  +  3aW— b*)=c-(a*— o)3 ; 

,     vTa2— B)<- 

hence  a? — b  = . 

c 

Since  a  and  6  ought  to  be  rational,  it  will  be  accessary  to  take  c  such  that 
(A3— B)c  be  an  entire  or  fractional  cube,  which  is  always  possible.  Calling 
the  second  member  of  the  above  equation  M.  we  shall  have  </-'  —  />=>!. 
whence  i=a3 — M.  By  substituting  this  value  of  b  in  equation  (3),  it  wiD 
become 


EQUATION  OF  THE  THIRD  DEGREE.  453 

4ca3— 3Mca— A=0 (4) 

This   equation  must  give  for  a  at  least  a  commensurable    -ulue,  without 
which  the  transformation  (1)  will  bo  impossible. 

If,  instead  of  VA-f-  \/B,  wo  should  have  to  reduce  y]  A  —  y/ti,  ir  would 
suffice  to  change  throughout  in  the  preceding  method  the  sign  of  V  b. 

'/  — 

For  example,  let  the  expression  be  \/14i  \/200.     We  shall  have  A  =  14, 

B=200,  A2— B  =  — 4;  hence  (A2— B)c=  —  4c ;  we  shall  then  have  the 
perfect  cube  —8,  by  taking  c=2.  Consequently,  M  =  —  1,  b  =  u-+],  and 
equation  (4)  becomes  8a34-Ga  — 14  =  0.  It  can  be  satisfied  by  the  commen- 
surable value  a  =  l,  which  gives  0=2.  Again,  we  have  already  obtained 
c=2;  hence,  finally, 


1/ 


14±  A/200  =  (l±  y/2)V>2. 


J  ZZ 

Again,  leu  the  expression  be  \  —  II  ±2  •/  —  1.     We  will  pass  2  und^r  tne 

radical  of  the  second  degree  ;  we  shall  then  have  A=  — 11,  B=— 4,  A2— B 
=  125.  As  125  is  already  the  cube  of  5,  it  will  suffice  to  make  c=l.  Con- 
sequently, we  have  M=5,  &=a'-— 5,  and  equation  (4)  becomes  4a3— 15a 
+  11=0.  But  this  equation  is  satisfied  by  the  value  a=l  ;  hence  fc=— 4, 
and,  consequently, 


V 


_ll±2V-l=(l±V-4)Vl. 


382.  Let  us  consider  the  more  general  expression  yAi  -\/B,  and  take 


J 


'A±  VB  =  (ai  Vb)Vc (5) 

The  problem,  again,  is  to  determine  rational  numbers  for  a,  6,  c,  if  it  be 
possible. 

Raising  (5)  to  the  power  n,  and  equating  separately  the  rational  parts,  we 

obtain 

n(n  — 1)        ,      n{n— 1)(»— 2)(n— 3)        .    ,     ,     ,  ,„, 

VB=C[na^+^^=^a»-»6+,&c.]V6   •  •  •  (7) 

We  can,  as  in  the  ca>e  of  the  cubic  radical,  square  these  two  equalities,  and 
subtract  the  one  from  the  other;  but  the  reductions  will  be  immediately  per- 
ceived by  observing  that  we  ought  to  have,  at  the  same  time, 

A+  VB=c(«+  Vb)*i  A—  VB=c{a—  V~b)n ; 
and  that,  consequently, 

A»— B=d>(a+  V~f>)n(a—  •v/"o)l,=cg(a8— 6)"  ; 

V(A8— B)c"-- 

whence  a- — b  = . 

•c 

We  see  from  this  that  it  will  be  necessary  to  take  c  of  such  a  value  that  the 
second  member  of  this  last  equation  shall  be  rational.  Calling  this  second 
member  M,  we  shall  have  a3 — 6=M,  whence  &=a*— M;  subsrirutin 


454  ALGEBRA. 

value  of  b  in  (6),  the  resulting  equation  in  a  will  have  a  commensurable  root 
every  time  that  the  transformation  (5)  is  possible. 

383.  In  the  resolution  of  equations  of  the  third  degree,  what  renders  the  ir- 
reducible case  so  remarkable  is,  that  although  we  are  assured  that  the  three 
roots  are  real,  it  is,  nevertheless,  impossible  to  make  the  imaginary  quantities 
disappear  otherwise  than  by  means  of  series.  This  difficulty  is  not  confined 
to  the  equation  of  the  third  degree  ;  it  will  be  encountered  equally  in  the  gen- 
eral formula 


VA+BV— 1  +  V^ 


'A-BV-l (8) 

which  formula  I  shall  stop  to  consider  for  a  moment. 

To  consider  this  expression  in  its  most  general  sense,  we  ought  to  comrjine 
the  n  determinations  of  the  first  part  with  the  n  determinations  of  the  second, 
so  that  we  shall  have,  in  all,  n-  values.  But  the  expression  is  rarely  taken  in 
so  general  a  sense,  and  I  proceed  to  define  that  which  we  ordinarily  attach 
to  it. 

As  the  two  radicals  which  have  the  index  n  represent  the  roots  of  the  bi 
nomial  equation,  their  determinations  are  equal  in  number  to  the  quantities 
which  have  the  form  f-\-g  V  —1.  Moreover,  it  is  manifest  that  to  each  de- 
termination of  the  first  radical  there  corresponds  one  of  the  second,  which 
only  differs  by  the  sign  of  yf —1.  But  we  suppose  that  these  corresponding 
values  are  those  which  ought  to  be  added  in  formula  (8) ;  and,  with  these  re- 
strictions, the  values  of  x  are  all  real,  and  only  n  in  number. 

The  product  of  these  two  radical  values,  thus  taken  in  a  same  pair,  is  real 
and  positive  ;  but  for  the  product  of  the  two  radicals  we  have,  in  general, 


Va+bv-ix\/a-bv-i  =  V 


VA^  +  B-\ 

and  the  radical  which  expresses  this  product  can  only  have  a  single  real  ana 
positive  value ;  hence,  if  we  represent  it  by  K-,  we  ought  to  be  able  to  charac- 
terize the  conjugate  values,  which  must  be  added  in  formula  (8),  by  the  con- 
dition that  their  product  be  equal  to  K2. 

Formula  (8)  can  be  regarded  as  a  general  expression  of  the  roots  of  an  equa- 
tion whose  degree  is  marked  by  the  number  of  values  of  which  the  equation 
is  susceptible  ;  hence,  provided  that  it  be  taken  in  its  greatest  extension,  oi 
with  the  restriction  which  wo  have  just  mentioned,  the  degree  of  the  equa- 
tion must  bo  either  n"  or  n. 

This  last  remark  leads  us  to  explain  how  we  form  an  equation,  when  we 
know  the  expression  for  its  root;  that  is  to  say,  thai  an  equation  being  given, 
susceptible  of  taking  different  values,  by  reason  of  the  multiple  values  of  the 

radicals  which  it  contains,  it  is  required  to  find  an  equation  free  from  radicals 
which  has  these  values  for  roots.     I  will  tako,  for  example,  the  same  expr 
sion  (8). 

To  abridge,  let  us  make 

A-rBV^T=(/,  ^.BV~l=5; 
the  problem  reduces  itself  to  eliminating  y  and  i  between  the  three  equation! 

y+z=.r,  >/  =  a,  v  =  h. 
But  hero  the  elimination  can  be  conducted  according  to  a  very  an  i\>    ,>i  o- 


EQ.UATION    OP    THE  FOURTH  DEGREIv  465 

cess,  analogous  to  that  which  has  been  employed  for  reciprocal  equations.     By 
the  rules  of  multiplication  we  have 

(r+~m)(2/+2)_=r+1+zm+1+2/-(r_^_+=m-1)- 

But  y-\-z=x  and  yz=%/ab  ;  hence,  making  y ab=c,  the  equation  will 
become 

yn-H  -\-zm+1  =x(ym  -\-zm) — c(?/m~I-f-;'"_1). 

By  means  of  this  formula  wo  express,  in  function  of  x  and  c,  successively  all 
the  quantities  y"-\-z2,  y3-\-z3,  &c.  When  we  have  arrived  at  yn-\-z",  we  re- 
place yn-\-zn  by  a-\-b,  and  then  we  shall  have' the  required  equation,  which 
will  be  of  the  degree  n  in  x. 

This  equation  contains  c  ;  but  we  have  c=  yah—  ?JA2-\-B2;  hence,  c  is, 
in  general,  susceptible  of  n  different  values.  By  putting  in  the  equation  each 
of  these  n  values  in  its  turn,  we  shall  have  n  equations,  and,  consequently. 
nXn,  or  n2  values  of  x.  This,  in  fact,  ought  to  be  the  case,  from  what  has 
been  said  at  the  close  of  the  preceding  article.  If  we  should  wish  to  have  a 
single  equation  which  has  all  these  values  for  roots,  it  would  be  still  necessary 
to  eliminate  c  between  the  equation  of  the  degree  n  in  x  and  the  equation 
cn=ab. 

But  if  in  formula  (8)  we  only  wish  to  associate  the  radical  values  whose 
product  is  real,  it  is  this  real  valuo  solely  which  we  must  choose  for  c,  and  we 
shall  only  have  a  single  equation  of  the  degree  n  for  determining  all  the  values 
of  x. 

RESOLUTION    OF   THE   EQUATION   OF   THE   FOURTH    DEGREE. 

384.  After  having  made  the  second  term  disappear,  the  general  equation  of 
the  4°  degree  is 

x4+2}x2+<lx+r=0 (!) 

If  we  make  x=a-\-b-\-c,  squaring,  there  results 

;r2=a2+&2-fc2+2(a&-f  ac+ic), 
)r,  transposing, 

a-2_(a»+Z.2+c2)=2(ai+flr+6c) ; 
raising  anew  to  the  square,  we  have 

;r*— 2(a2+&2+c:):r2+(a2-}-^+e-2)2^^ 
then,  replacing  a-{-b-\-c  by  x,  and  transposing,  we  obtain 

.r4_2(a24-62+c2).r2— 8rtic.r+(a2-j-?,2-f  c2)2 
_  4  (a-b2  +  a2c2  -f  b-d2) = 0 . 

This  equation  is  without  a  second  term,  and  by  the  manner  in  which  it  has 
been  formed,  we  know  that  it  admits  of  the  root  x=a-\-b-\-c.  Thus,  we  re- 
solve equation  (1)  in  determining  a,  b,  c,  by  the  condition  that  it  shall  hi  iden- 
tical with  the  preceding,  which  gives 

—  2{a"+b-  +  c2)=;> 
—  8abc  =  q 
(a2-f  62+c2)2— 4(a262+a2c2+i-c-)  =  r. 

These  equalities  show  that,  by  taking  a2,  62,  c"  for  unknowns,  these  three 
quantities  are  the  roots  of  an  equation  of  the  3°  degree,  the  coefficients  of 
which  are  (see  Art.  245) 


456  ALGEBRA. 


_(a»+6»+c»)=:| 


2 

p2— 4r 


•  lo 

9* 
64 

Consequently,  this  equation  of  the  3°  degree  is 
p         ■))"- — 4r        q" 

+  2~  +      16    ■-      64_U W 

Such  is  the  reduced  equation  upon  which  the  solution  of  equation  (1)  depends. 
Suppose  that  the  three  values  of  z  have  been  determined,  which  designate 
by  z',  z",  z'",  we  shall  have 

«=±  V*,  6=±  V^77,  c=±  V~'- 
If  the  signs  be  combined  in  all  possible  ways,  there  will  result  eight  values 
for  a-\-b-\-c  or  x.     But  as  the  last  term  of  the  reduced  equation  (2)   was 

formed  by  squaring  the  equation  abc= —  -q,  it  follows  that  the  values  contain 

o 

rtot  only  the  roots  of  the  proposed  equation,  but  also  those  of  an  equation 
which  would  differ  from  it  in  the  sign  of  q. 

At  the  same  time  it  may  be  perceived  that,  to  have  only  the  roots  of  the 

proposed,  it  is  nocessaiy  to  add  only  the  values  of  a,  b,  c,  for  which  abc=  —  -q, 

8 

and  the  product  of  which  has,  consequently,  the  contrary  sign  to  q.     In  each 

particular  case  it  will  be  easy  to  determine  for  the  radicals  three  values,  A,  B, 

C,  which  shall  fulfill  this  condition ;    and  afterward,  with  these  values,  we 

form  the  four  roots  of  ihe  proposed,  to  wit, 

.r=  +  A+B  +  C,  .t=  +  A— B-C, 

x=  —  A+B  —  C,  x=—  A  —  B  +  C. 
Generally,  instead  of  A,  B,  C,  the  three  radicals  are  placed,  and  the  values 
of  x  are  written  Jhus  : 

x=  +  ■/£+  V^-  V£5  x=  +  -/£-  V£+  v5f . 

x=-  yfz'+  V~"+  Vz'",  x=—^z'-  Vz"-  V-'". 
But  it  is  necessary  to  understand  that  in  applying  these  formulas  to  particu- 
lar cases  there  must  bo  taken  for  -\/z',   -J z",  yf ':'"  three  determinations,  tho 
product  of  which  shall  be  of  tho  88  n  as  q.     This  observation  is  im 

portant;  failing  to  have  regard  to  it,  we  might  find  false  ro 

385.  The  nature  of  the  roots  of  the  reduced  equation  will  make  known  the 
nature  of  tho  roots  of  tho  proposed.     But  tho  reduced  ha\  last  term 

negative,  has  always  one  positive  mot  (see  \rt.  '.'!-.  Prop.  VIII.,  Cor.  4),  and 
the  product  of  tho  other  two  roots  should  bo  positive  ;  then,  if  these  last  are 
not  imaginary,  they  will  be  both  positive  or  both  Degative.  I  pass  over  the 
case  in  which  q=0,  because  then  the  proposed  would  b  -  solved  by  the  rules  foi 
the  second  degree.    Consequently,  there  are  three  cases  only  to  be  examined.  * 

1°.   Cast  where  the  three  roots  of  the  reduced  equation  arc  positive.     There 

the  four  values  of  .(•  are  evidently  real,  and  if  the  radical 

regarded  as  representing  positive  determinations,  their  pro  luct  will  be  positive; 


This  explains  an  operation  in  Art 


DIOPHANTINE  ANALYSIS.  457 

then  tn«  preceding  formulas  will  bo  specially  applicable  to  the  case  of  <7^>0. 
For  q<^0  it  would  be  necessary  to  change  the  sign  of  one  of  the  radicals. 

2°.  Case  where  the  reduced  has  one  root  z'  positive,  and  two  z",  z'"  negative. 
The  radical  Vz'  will  be  real,  but  the  radicals  V '  z"  and  -<Jz'"  will  be  imagi- 
nary ;  consequently,  the  four  values  of  X  will  be  imaginary  also,  unless  2"=;'". 
When  z"=z'",one  of  the  two  quantities  V~"+  y/z777  and  y/~  —  -/p7  will 
become  zero,  and  supposing  it  to  be  the  latter,  the  values  of  x  will  be  6imply 

x=  -v/z7,  x  =  -/I77,  x=  —  •v/I7+2  y/U7,  x=  —  V~'—2  Vz"- 

The  first  two  are  real,  since  z'  is  positive,  and  the  other  two  avo  imagi- 

naiy,  since  z"  is  negative.     Besides,  as  in  the  reduction,  wo  have  supposed 

■\/z"=  y/-z'",  we  ought  to  have  here   y/z'  \J  z"  y/z'"=z"  y/z' ;  so  that  tins 

product  can  only  have  the  sign  of  q  by  choosing  for  y/ z'  a  sign  contrary  tc 

that  of  q,  since,  by  hypothesis,  z"  is  negative. 

3°.  Case  in  which  the  reduced  has  one  root  z'  positive,  and  two  roots  ■/.",  z'  ' 
imaginary.  The  positive  root  z'  being  known,  we  can  divide  the  reduced  by 
x — z',  and  wo  shall  have  an  equation  of  the  second  degree,  which  will  give  for 
z"  and  z'"  imaginary  values  of  the  form 

z"=f+g  V=l,  z'"=f-g  /="l. 
Consequently,  two  of  the  values  of  .r  will  contain  the  sum 

yjf+g  /ZI+V/-«  V"^l ; 

and  the  other  two  will  contain  the  difference 


Vf+gV-i-\lf-gV-i- 


THE  DIOPHANTINE  ANALYSIS. 

386.  This  branch  of  analysis  derives  its  name  from  its  inventor,  Diophan- 
tus,  of  Alexandria,  in  Egypt,  who  flourished  about  the  year  360,  A.D.  It 
relates  chiefly  to  the  finding  of  square  and  cube  numbers. 

The  solutions  of  the  questions  must  frequently  be  left,  notwithstanding  the 
various  rules  that  have  been  given  for  this  purpose,  to  the  talents  and  ingenui- 
ty of  the  learner,  who,  in  pursuing  these  inquiries,  will  soon  perceive  that 
nothing  less  than  the  most  refined  algebra,  applied  with  great  skill  and  judg- 
it,  can  surmount  the  various  diff  which  attend  them  ;  and,  in  this 

respect,  no  one,  perhaps,  has  ever  excelled  Diophantus,  or  discovered  a  greater 
knowledge  of  the  extent  and  resources  of  the  analytic  art. 

When  we  consider  his  work  with  attention,  we  are  at  a  loss  which  to  ad- 
mire most,  his  singular  sagacity,  and  the  peculiar  artifices  he  employs  in  form- 
ing such  positions  as  the  nature  of  the  problems  requires,  or  the  more  than 
ordinary  subtilty  of  his  reasoning  upon  them. 

Every  particular  question  puts  us  upon  a  new  way  of  thinking,  and  fur- 
nishes a  fresh  vein  of  analytical  treasure,  which  can  not  but  prove  highly  use- 
ful to  the  mind  in  conducting  it  through  other  difficulties  of  this  kind  when- 
ever they  occur,  and  also  in  enabling  it  to  encounter  more  readily  those  that 
may  arise  in  subjects  of  a  different  nature. 

The  following  directions  for  resolving  questions  in  the  Diophantine  analysis 


458  ALGEBRA. 

will  be  found  useful ;  but  no  general  rule  can  be  given,  and,  therefore,  tne 
student  must  often  be  loft  to  depend  solely  upon  his  own  ingenuity  and  skill. 

\ 

RULE. 

Substitute  for  the  root  of  the  square  or  cube  required,  one  or  more  letters, 
such,  that,  when  they  are  involved,  either  the  given  number  or  the  highest 
power  of  the  unknown  quantity  may  vanish  from  the  equation  :  and  then,  it' 
the  unknown  quantity  bo  of  the  first  degree,  the  problem  will  be  solved  l»y 
reducing  the  equation.  But  if  the  unknown  quantity  be  still  a  square  or  a 
higher  power,  some  other  new  letters  must  be  assumed  to  denote  the  root, 
with  which  proceed  as  before,  and  so  on  till  the  unknown  quantity  is  but  of  the 
Irst  degree,  and  from  this  all  the  rest  will  be  determined. 

EXAMPLES.* 

(1)  To  find  two  square  numbers  whose  sum  is  a  square. 
Let  x"  and  y-  be  the  two  squares  ;  let  3z  and  4;  be  the  roots. 


Then  25:2=  D  f=n  — bz\-  =  n°~— 10«;  +  25z2; 

.-.z=— —  ;  if  n  =  10,  z=l,  then  ;X3  =  3  and  :X4=4  , 

and  the  two  squares  are  9  and  1G,  whose  sum  is  25,  a  square,  if  n  =  20,  :=2 ; 
and  from  this  we  get  another  value  of  x  and  y,  and  so  on. 

(2)  To  find  two  square  numbers  whose  difference  is  a  square. 
Let  x-  and  y-  be  the  two  squares. 

Assume  a:2 — y2=:{x — ny)~=x" — 2nxy-\-n"y". 
Then  —if=—2nxy+n°tf-, 

or  2nx=(n7-\-l).y  ; 

n2-4-l 

•■•  x=-n — -y- 

2/i      J 
Suppose  y=2n,  then  x=n"-\-l.     If  n=2,  y=l,  and  x=5  ;  also  J8 — y3 
=  25  — 16=9,  a  square  number.     If  n=3,  y  =  G.   and  .r=10;  also  x* — y9 
=  100 — 36=G4,  a  square  number. 

(3)  To  change  the  sum  of  two  squares  into  the  sum  of  two  others  any  num- 
Der  of  ways  at  pleasure;  for  example,  in  three  different  ways. 

Let  a"  and  6s  be  the  given  squares,  and  let  a — x  and  ex  —  b  be  the  roots  of 
the  required  squares;  then,  by  the  question,  we  get 

a—x\■z+'cx^l>\:  =  a-  +  L^3■, 
by  involution,  a2— 2a.r-|-.r2-f.c2.r2  — 2b<\r+lr  =  u ■  +  /.■  ; 

by  transposing  and  dividing, 

_ort_j_.r_j_  C<!X_ o(7,_o, 

26c+2a 

cir  c<ix4-x=2bc-lr-2,i  and  .r=  — — — — , 

1~T" 

where  c  may  bo  taken  at  pleasure  ;  for  example, 

c  =  2,  3,  and   I  , 

4o-|-2a   66+2a  Bo+2a 

then,  '= — ,  — — — ,  and  — — — . 


*  Many  of  these  problem!  are  sell  cted  from  tin-  Arithmi  deal  Questions  of  Diopbantos, 
of  which  six  out  of  tin'  ow  remain.    The  best  edition  is  that  published  at 

Paris,  bj  /■'  "  het,  in  the  year  1670,  with  notes  by  Fern 

t  This  sign  □  denotes  that  the  number  placed  equal  to  ii  is  a  perfect  scjuaru 


DIOPIIANTINE  ANALYSIS.  459 

(4)  To  divide  a  number  which  is  the  product  of  the  sum  of  two  squares  oy 
the  sura  of  two  others,  into  two  squares  two  different  ways. 

Let  a"-\-l~  bo  tho  sum  of  two  squares,  and  c2+rf2  the  sum  of  two  others, 
whose  product  (a2+&2) .  (c'i+di)  =  {ac  +  bdy+(bc—ad)*={ac—bd)-  +  (6c 
•\-ady,  as  required. 

(5)  To  find  a  number, .r, such, that  .r+1  and  x— 1  shall  bo  squares. 

Let 
and 


or 


and 


x+l  = 
x  — 1  = 

=  b* 

.-.  2= 

=a*- 

-b*  by 

subtraction ; 

.-.2X1  = 

3: 

=  (a+b)(a- 
=2a,  and  a 

3 

=2' 

a2 

9 
-4; 

.-.  .r+1 

9 

=  4' 

and  x= 

5 

"4' 

x+1 

=f 

x 

=f  —  \ 

Or  thus 


x — 1=2/2 — 2=   □   — s — ?/|-=6-c — 2s2/+3/2 

.-.  s2  — 2sy  =  —  2 

s2+2 
2st/=s2+2,  andy=-£- ; 

9  5 

take  s  =  l  •••  2/=i>  anc^  z=^3 —  1=7 —  1=:I'  as  before. 

(6)  Required  to  find  four  square  numbers  whose  sum  shall  be  a  Q  . 

Let  1,  4,  9,  and  x2  be  the  required  squares;  then,  by  the  question,  we  get 

14+x2=  D   =7i—  x|3=n2— 2nx+x2, 
n-  — 14 
and  x=—2n- 

5  25 

where  n  may  bo  any  number  at  pleasure,  if  w  =  3,  .r= — p  x2=— ,  or  if  n=4, 

1  11        225    ~15 

x=-,  and  the  numbers  are  1,  4,  9,  and  —  ;  then  1  +  4+9+—  =  —-=— 

as  required. 

(7)  Divide  2  into  three  rational  squares. 
Let  x,  2.r — 1,  and  3x — 1  be  the  roots  of  the  three  squares  respectively: 

then  x2+4x2— l.r+l  +  9x2— 6x+l=2 ; 

by  transposing  and  dividing, 

5  3  8 


x=-,  2r — 1  =  7'  3x — 1=7>  the  roots; 


and  the  Q  's  will  be 


25  ,        9         , .       64 

*2=49'  2a;-1l2=i9'  anl  3^~ li3=49' 

„     ,     ,       25      9      64      98 
the  sum  of  which  is  -  +  TJ^+TJj^  — ~  the  Proof- 

Or  thus : 

Let  1,  x2,  and  y-  be  the  squares ;  then 


46G  ALGEB&A. 

1-f  r2+?/-=2  and  ^+i/:  =  l, 
or  x2 = ]  —  ?/2  =  D  =  1  — ny\2  =  1  —  ~>nj  -J-  "'32/* 

2rc 

4 
where  n  may  bo  taken  any  number  greater  than  1  ;  if  n  =  2,  then  y=   -   and 

16  9 

y2=—  ;  then  will  x2=— ,  and  the  sum  of  these  plus  1  is  evidently  2. 

(8)  Divide  -  into  three  rational  squares. 

Let  x,  2x — -,  and  3x — -,  be  tho  roots  of  the  rational  squares,  and  then 

squares  are 

1  1 

x2,  4x2— 2x+-,  9X2— 3x+-, 

and  x2+4x2— 2x4-7+9x2  — 3.r+-=-, 

i  141  '42 

5  25 

and  x  will  be  found  to  be  — ,  from  which  we  get  the  three  squares,  viz.,  ~, 

9      G4  1 

77^-,  tt^i  a»d  their  sum  is  evidently  -,  as  required. 
19o   Ufa  2 

(9)  To  divide  a  given  square  number,  100,  into  two  such  parts,  that  each 
of  them  may  be  a  square  number. 

Let  x2  be  one  of  the  parts,  then  100 — x2,  the  other  part,  will  be  a  square 

number. 

Assume  100— x2=(2x— 10)2=4x2 — 10x-flb0. 

.*.  x=8,  and  2x — 10  =  6 ;  honce  64  and  36  are  the  parts  requin  J. 

The  same  problem  may  be  resolved  generally  in  the  following  manner : 

Let  a2  be  the  given  square,  x2=  one  of  its  parts,  and  a2 — x2  the  other. 

Assume  a2 — x2  =  (nx — a)2=n2x2 — 2anx-\-cP\ 

Then  — x2 = ;t:x2 — 2a«x  ; 

2na  an2 — a 

-,  and  nx —  a: 


•■-- „*+l'-""  "*      "-n=+l   ' 
/  2na  y  lan2—a\2 


2-1-1/        \ri--\-] 

are  the  two  squares  required  ;  in  which  expressions  a  and  n  may  be  any  whole 
numbers  whatever,  provided  n  be  greater  than  unity. 

(10)  To  liud  a  numboiyr.sucb.that  x-j-128  and  x+192  shall  bo  both  square 
numbers. 

Assume  x-f- j.28  =  ::  •*•  £=  -  —  1  '-' 3,  which  is  one  condition  answered;  then 
2s— 128+192r=zs+64=E  D  =a2  .■.z2=a-— 64  ;  then  wo  have  only  to  assume 
such  a  value  for  a  as  will  make  <c  —  04  a  square;  but  it  is  plain  that  if  a  be 
taken  =10,  then  a2—  -64=36=  □  ,  and  ;2  =  36;  but  this  would  make  the  value 
of  x  negative,  then,  in  order  to  find  values  for  :  that  will  make  X  positive,  take 
a  =  17,  and  the.i  «*=289,  and  •••  0*—  fa  1=225=  □  .-.  :-'  =  225  and  •••  s  =  -22o 
— 128=97,  the  value  required. 

(11)  To  divide  a  given  Dumber,  1 3,  consisting  of  two  known  squares,  9  and 
4,  into  two  other  square  numbers.* 

*  in  the  eolation  given  of  the  above  problem,  n  awl  m  may  be  taken  emu]  to  any  nuin 


DIOPHANTINE  ANALYSIS.  4G1 

Let  nx — 3  be  the  root  of  the  first  squaro  sought,  and  mx — 2  the  root  of 
the  other  squaro. 

Then  nx—3\2-\-7nx—2\2=13, 

or  (n3-fm2)  .  z2=(4wi  +  6?i)  .  x; 

6n-\-4m 


x  = 


n-^-m 


2    ' 


3n2+4?;m— 3m2 

whence  nx — 3= — — =  the  root  of  the  first  squire. 

n--f-m 
6mn— 2n2+2m2 

and  mx — 2= — — - =  the  root  of  the  second. 

?i2+m2 

„       ,                    ,         3ra2+4m«— 3m2     17 
It  n=2  and  m=l,  we  have — — 5 =— =  the  rootot  one  square, 

(§m— 2rc2+2m2     6 
and — — ^ =7=  tne  ro°t  of  the  other  square. 

(12)  Let  14  bo  divided  into  three  rational  squares.  It  is  well  known  that 
the  least  three  squares  in  whole  numbers  are  1,  4,  and  9,  which  will  answer 
the  question;  but  to  give  a  general  solution, 

Let  1,  3x — 2,  2x — 3,  be  the  roots  of  the  required  squares; 

24 
then  l  +  (3.r— 2)2+(2z— 3)2=14,  or.r=— ; 

24  72   „ 

then  — X3  =  — ,  from  which  subtract  2  ; 

J.  O  J.  O 

/46\2     2116     24  48 

then  y— J  =-^n  »  T3  X  2=i3'  "'om  wllicl1  subtract  3  ; 

/9\2      81  2116      81 

then  {-)  =^  .-.  1+  — +_=14. 

(13)  To  find  two  square  numbers  whose  difference  shall  be  equal  to  any 
given  number. 

Let  x  be  the  root  of  the  lesser  square  sought ;  and  let  d,  the  given  difference 
of  the  squares,  bo  resolved  into  any  two  unequal  factors  a  and  b,  of  which  a  is 
the  greater. 

Let  x-{-  b  be  the  root  of  the  greater  square ; 

then  (x-{-b)a — xn-=d=ab, 

i.  c,  2x-\-b=a, 

a — b 
Whence  x=— - — =  the  root  of  the  lesser  □, 

a  +  b 
and  a:-f-&  =  — — =  the  root  of  the  greater. 

Tf  d=60,  and  a  X  6=30  X  2,  we  have 

30—2  30+2 

— — =14,  and— J-^=16; 

whence  162  and  142  are  the  squares  required  whose  difference  =60. 

bers  whatever,  provided  their  ratio  be  not  that  of  3  :  2.    For  if  n  were  to  to  as  3  to  2,  the 
roots  of  the  squares  sought  would  be  found  the  same  as  the  roots  of  the  known  squares. 
Tf  it  were  required  to  divide  a  given  square,  ofi,  into  twe  other  squares, 

Since  {m*-\-n")-={m°—  n"-)--\-{fimn)^, 

.-.  (m3_J_¥jj)s  .  xi—{rrfl—ivi)i  .  ^-(-(2/»kVJ  .  x-2, 

where  m  and  n  may  be  assumed  at  pleasure,  m  being  greater  than  n. 


4b'2  ALGEBRA. 

(14)  To  find  two  numbers,  such,  that  if  either  of  them  be  added  to  the 
square  of  the  other,  the  sum  .shall  be  a  square  number. 

Let  x--\-2xy  and  y  bo  the  required  numbers  ; 
then  x2+2xy-\-if=  D  =x-lry\%  ; 

hence  it  only  remains  to  make 

2/-j-2.n/+j.-'p=  U  =xi-\-ny\-=x*+2nx2y-\-niy*, 

l^_4.,-3_2«.r!J 

•"•  y=      w2— 4x2      ' 

3        ,  19 

If  n=2j,  and  z=l,  then  2/=r^  and  £2-|-2.n/=— ,  which  are  two  numbers 

that  will  answer  the  conditions  ;  for 

Y 

13 


2      19      256      16 
+  I3  =  169=r3 


19 
and  13 


'       3  _400_20|2 


13      169      13 


Or  thus : 


Put  - — x  and  x  for  the  numbers;  then  - — x-|-x2=(- — x)  ,  a  square,  and 

=  D  ,  where  .r  may  be  taken 


1  1      .t  1      a:  1 

+X=16-5+3:+-T-i6  +  o4-2=4+.r 


4-r 

at  pleasure,  provided  it  be  less  than  —. 

(15)  To  find  two  numbers  whose  sum  and  difference  shall  be  both  square 

numbers. 

Let  x  and  y  be  the  two  numbers  ;  then,  by  the  question, 

x-\-y=  o  =a2  and  x — y=.  d  =is ; 

add  both  squares,  and  we  get 

2or=a2+62; 

a24-i3 
hence  x= — - — . 

Again,  by  subtraction, 

2y=a2 —  62  and  y= — - — , 

where  a  and  b  may  be  taken  at  pleasure,  provided  a  be  greater  than  b ;  if 

9  +  1  9  —  1 

a  =  3  and  b  =  l,  then  — - — =5  and  — -—=4,  whose  sum  and  difference  are 

both  squares.     Or  thus  : 

Let  x  and  x* — x  be  the  numbers. 

It  is  evident  that  their  sum  is  a  square  ;  and,  in  order  to  satisfy  the  other 
condition  in  the  question, 
Assume  x — n|2=x2 — 2x,  the  difference  of  the  numbers  ; 

whence 


2n— 2' 


.-.  X  —X=  ^  2n_2  |     —  o;i_o' 

«2  i      n2      )  2         n2 

Hence  the  two  numbors  aro  -— iuid   < >   — -,  ia  which  n  inav 

2n  —  2  {  2n  —  2  )         2n  —  2 

9 
bo  taken  at  pleasure,  provided  it  be  greater  than  1.     If  7i  =  3,  J=t,  and 

45 

x° x . 

16 


DIOPHANTINE  ANALYSIS. 


4G3 


^16)  Find  two  numbers  Whose  sum  is  a  square,  the  sum  of  their  squares  a 
square,  and  either  added  to  the  square  of  the  other  a  square. 

Lot  7 — x  and  x  bo  the  numbers  ;  then  their  sum  7  is  a  sauare,  and  - — x 
4  4  4 


+x2 


D=--x 


1        x  1         » 

a  square,  and  —  — --{-x-|-x2=  u  =--|-x    a  square  ;  and, 

in  order  to  satisfy  the  other  condition,  we  assume 

1      x  1 


nx      1 


n- 


•  1  3  14  4 

which,   solved,    gives   x=77j^,  if  ra=4,  x=— ,  and  7— x=— ,  so  that  — 

3 
and  — -  are  numbers  that  answer  the  condition?  as  foUows  : 

.CO 


3 

28 


+ 


28 


25     __5_ 

:^=28 


and2l 


'■      4        9         112        121        1113 

+  28— ^p+-fSl2— =]3— 28    5 


28 


also, 


4 
28 


i      3       16        84        100        10 

1  oq      msirr;!       7r^~,i      oq 


28     28l2     281 


28 


;js 


(17)  Find  two  such  numbers,  that  if  their  product  be  added  to  the  sum  of 
their  squares,  the  sum  shall  be  a  square. 

Let  2x  be  their  sum  and  2y  be  their  difference ;  then  the  greater  will  be 

r-\-y  and  the  less  x — y ;  hence  x" — y"  =  their  product,  and  2x2-f-2;y2=  the 

sum  of  their   squares;  then,   by  the   question,  3x2-{-y-z=  □  =nx — y\2   and 

2n  y 

x= : —  ;  if  n=2  and  V=2,  .  .  x=8,  which  will  answer  the  conditions. 

n2 — 3 

(18)  To  find  two  square  numbers,  such,  that  the  difference  of  their  cube 
roots  shall  be  a  square  number. 

Let  x6  and  y6  be  the  required  numbers.  Then  X? — y"=  O  ;  consequently,  X 
and  y  may  be  any  two  numbers  which  are  the  hypotenuse  and  one  leg  of  a 
right-angled  triangle,  and  the  least  lumbers  of  this  description  are  5  and  3,  and 
the  numbers  themselves  15625  =  1252  and  729=273. 

(19)  Find  three  numbers,  such, that  not  only  the  sum  of  all  three  of  them, 
but  also  the  sum  of  every  two,  shall  be  a  D  . 

Put  4x,  x2 — 4x,  and  2.r+l  for  the  three  numbers  ;  then  it  only  remains  to 
render  6x-|-l  =  □  • 
Assume  its  root  n  —  1  ; 

then 


whence 


6x+l=?i  —  ll2=n,2— 2»+l; 

X= : 


■xn 


if  ns=12,  x=2^,  which  will  answer  the  conditions  of  the  problem. 

(20)  Find  two  numbers,  such,  that  the  sum  of  their  squares  and  the  sum  of 
their  cubes  shall  be  both  squares. 

Let  b  be  the  base,  p  the  perpendicular,  and  h  the  hypotenuse  of  a  rational 

right-angled  triangle,  x  any  multiplier  of  b,p,  and  h  ;  then  (bx)--{-  (px)2=(hxy, 

but    (bx)z-\-(px)3=    a    rational    square    =r\r2 ;    hence    (b3-{-p3).x=ri,    01 

r" 
x=,3  ,    ■; ;  now  if  r=b3-\-p5,  .  .  xss&'-j-p3,  and  .-.  bx=b(b3-\-p3),  px=py 


464  ALGEBRA 

(bi+f),  now  let  6  =  3,  p  =  i;  then  is  x=(Jl,  6x=273,  a 
6=6  and  p=8,  then  x=728,  6r=4368,  and  pr=5824,  and  so  on 

(21)  Find  a  number  to  which  if  8  be  added,  the  sum  shall  be  a  cube.  Bud 
from  which  if  1  be  subtracted,  the  remainder  shall  be  a  cube. 

Let  x  be  the  number;  6=2,  esl;  then  r+6"  =  a  cube  and  x—?=  a 
cube  ; 

hence  •+63=(6  +  ^03=63+3c2a  +  -£r«2+^3 : 

Assume   x— c*  =  {a— c)»=   a    cube    =a3— 3a2c+3ac2  — c\    and   .-.  x=a> 

_9a2c_j_3ac2;  and,  equating  both  values  of  x,  we  get 

3c4         c6 
a3— 3a3c+3a^=3ac2+-^-a2+^-6a3, 

63+c3         7         3c63 
whence  a==66— c*X3c        63— c3 ' 

■ad,  putting  the  right-hand  member  of  this  equation  into  numbers,  we  get 

3X8     24 

a-8-l~7  '' 

5256 
hence  x — "343"' 

(22)  To  find  three  square  numbers,such,  that  the  sum  of  every  two  of  thera 
«hall  be  a  square  number. 

Let  x2,  y\  and  z2  be  the  numbers  sought. 

Then  x2+z2,  2/2+z2»  and  x2+t/2  are  the  three  numbers ;  i.  e., 

X1  yi  X1  y* 

-+1,^+1,  and -+^ 

are  threo  square  numbers. 

x     m2 — 1  y     n2  —  1 

Assume  7=  -gjp  *nd  -  =— , 

we  have 

x2  m-«_i-2m2+l        ,  y8  »4+2n8+l 

-:4-l  = r- ; )  anc»  ~7+1:= n. < 

22~  4»i2  =2  '  4»8 

x?+i/J 

which  are  evidently  two  squares;  and  therefore  it  remains  to  make  — ~r~a 

square  number. 

Now 

x2+y8_  (  m2— 1  \  2       i  w8— 1  )  -      (w8  — l)a     (n8— 1)3_ 
~~?"~— {"l^Tf    "M     2«     J     ~     4m2       +      4n2 
(ot°— l)3.ng+(w8  — l)8.m3 
4m27i3 
a  square  number. 

Hence 
(ro2  —  l)».n9+(n«— l)2.m2,  or  (>«  +  l)2.  (m-l)2.7i84-(w+l)2.  (n  —  ])'.  m»=a 
a  square  number. 

Let  m-\-\z=.n  —  1  .-.  7i  =  m-j-2. 

Hence        (wi+1)2.  (m  —  l)-' .(»' -+--)•  +  '"• ' 


DIOPHANTINE  ANALYSIS.  165 

or  (m  —  1)s  •  (?h  +  2)2-|-to2  .  (m+3)2, 

or  2>n4+8m3-\-6m2— 4m  +  4, 

is  a  square  number. 

,      5m2 
Let  the  root  of  this  quantity  be  assumcd  =  — —  —  7n-\-2. 

/5m2  \2 

Then  (— — m+2j  =2wi4+8m3+Gm2— 4/re+4 

whence  m  =  — 24,  and  n  =  — 22. 

x     m2— 1       575  y     n-— 1       483 

Also,  tsss— - = — — ,  and  -=— -z — = — —  ; 

z        2m        — 48  z        2n        —44 

575:        ,  483« 

henco  x= —    —  ,  and  y= — — — -. 

48  J  44 

To  obtain  the  answer  in  wftolo  numbers,  let  2=528  ;*  then  r=  — 6325,  and 

y  = — 5796.     Hence  528,  — 5796,  — 6325  are  the  roots  of  the  squares,  and 

5282,  579b2,  63252  are  the  squares  required. 

(23)  To  find  three  cube  numbers,  such,  that  if  from  every  one  of  them  a 

given  numoor  1,  be  subtracted,  the  sum  of  the  remainders  shall  be  a  square. 

Let  l-f-.r,  2 — r,  and  2  represent  the  required  roots. 

Then,  per  question,  (l-f-.r)3  — 1  +  (2— xf  — 1  +  8  —  1=  a  ; 

or  (l+x)34-(2— .r)3+8— 3  =  a  . 

ar5+3x-2+3.r-j-l  +  8  — 12.r+6.r2— .x^+S— 3=  D  ; 

9x2—9x+U  =  a  ,  =(a— 3x)2=a2— 6ax+9x*  ; 

11  —  9a-=a2— Gax; 

a2— 14 
and  6ax — 9.r=a2  — 14  .•.  .r= 


6a— 9 
_17 
15  T~X— 15'"  "15' 


16-14      2  17  28 

Suppose  rt=4  ;  tuen  x= — — — =Tt>  an"  l+-r:=T7>  anc*  ~ — x= 


4913    ,  /28\3     21952 

•••  <*+*>  -3375'  &-X)  -U  ="337^  and  8 
are  the  numbers. 

(24)  It  is  required  to  find  three  integral  square  numbers,  such, that  the  dif- 
ference of  every  two  of  them  shall' be  a  square  number. 
Let  the  roots  of  the  required  numbers  be  denoted  by 

s-+tf<,  s2-y2,  and  r2+x2. 
Assume  r2 — x-=s--\-y-; 

then  r-  —  x2—s-=y-=  O 

and  y* = r* — 2>--x-  —  2 r2s- -\-  x* -\-  2x"s'  -f-  s4 ; 

but  (f2+a«)a— (ss —y-)-  =  D 

=  (r2+x2)2  —  (s2  —  r°~+x2+s2)2=ri  +  2r2x2+x4  — s*+2r-s-— 2s"-x2  —  2s*— *• 
+  2r2x2 + 2r2s2 — x4 — 2s2x2 — s4  =  □ 
=4r2x2+4?-V— 4s2.r2— 4s4=  D 
= 4 (r2.r2 + rV — s-.r2 — s4)  =  D  ,  # 

.-.  r2x^-{-r2s2—s"'x2—s4=  a  =a2, 
and  (r2 — 52)  •  x"- = a2  —  r2s2  -f  s4, 

flS_rV+«<  fl3 

and  2?= r : —  =  — -—s2 ; 

r-  —  S^  7-  —  S2 

take  r=21  and  s  =  13, 

*  The  least  coirunon*  multiple  of  the  denominators,  49  and  44. 

G  G 


466  ALGEBRA. 


r- 


**n  -2=i4T^l69-169 


a 


2 


Take  a  =  340, 

then  a^=25G  and  y"=r2— s-— x2=441  — 25G  — 169  =  :  6, 

.-.  (/---f.r-)-  =  (441-r-256)2=(697)2  =  one  numter, 
and     (r3— *3)2  =  (s2+2/2)2  =  (441  — 25G)2  =  (185)2  =  the  second  number, 
and  (s2 — ?/2)2=(169 — 1G)2  =  (153)-,  which  is  the  other  number. 

(25)  To  find  three  square  numbers  such,  that  their  sum,  being  severally 
added  to  their  three  roots,  shall  make  square  numbers. 

Let  2t,*6x,  and  9.r  denote  the  three  roots;  .-.  by  the  question, 

121z2+2.r=D,   * 
121x2+6.r=  d. 
121z3-r-9.r=  D- 

Assume  x=-^- ;  then  121x=y ;  and  .-.  121.r-=-^-,  and  121xs+2.r=^- 

+121' 

Hence,  we  got 

2/24-2?y=  D, 
2/2+G2/=n, 
2/2+9t/=D. 

(Z'  —  \Y  ,  /23  — 1\-  Z*— 2z«+l 

\Ssume2/2+23/=^-5rj  ;  and  .-.  if +2^1  =  [-^-j  +1= ^3- 

2«_2z3+4^4-l      z4+2z2+l       /2J+1\2 
+  1= — = — =y-^~J   ;    and,    consequently,   7/-fl 

=--|-l  zs+l  z2  —  22+1      (2  —  l)s 

=— - —  .-.  =y=— — — 1= = — ;  houce,  by  substitution  in  the 

22  ^z  ' ' "  Lc 

second  equation  above,  wo  have 

(z  —  lY  {z  —  l)a  (2  —  1)<  (:-l)2 

Rut  422  is  a  square  number ; 

...  (z—l)«+12zx(z— l)a=D 
=  (2-l)2x(^-l)2+122.(^-l)2  =  (r-l)=x|(2-l)2+12r|. 
But  (2  —  l)2  is  n, 

...  (z— 1)»+12*=  D  =22+102  +  l=  a  . 

4gain,  by  substitution  in  the  third,  wo  have 

^=^4-9 x^=^-  D      (*-l)y8*X(z-l)» 

42a    -rJ*      or     — u—    4. j    "t"         4,3         —  Ui 

..  (2-l)'+183X(:-l):=a,aud.-.  (c-l)-\(2-l)2-fl8:.(:-l)«=a 

Hence  (:— l)2X  {(2  — 1)3+18:  j  =  d, 

and  .-.  (2  —  1)2+182=D  =::+lC:  +  l  ; 

hence  (z»+16z+l)—  (::+10r-fl)  =  G:  =  :::  X  -1. 

,1  3z+2 

the  -  sum  of  which  factors  is  — - — =—  +  1,  the  root  of  the  greater  Q . 


/3z        \« 
*»+16z+l  =  ^+l)  =  T+3:  +  l, 


ana 


DIOPHANTINE  ANALYSIS.  467 

Or2 
«»+16*=— +  3i,  and  422+64z=9,z2+122, 

52 
.-.  4z+64  =  9z+12,  5z=52,  and  ?=—  ; 

o 


/52        \2      /47\2     22 


09     2209 


,5        /         \5/  25         25 

•2/=- 


52  52  52 

~X"7~  ~X  "T"       ~""c" 


15 


5 

_2209  y       _  2209 

—"520  a'ld  *-"121'  ~"  62920  ; 
4418     13254        ,  19881 
•'   We  SC6  that         62920'  62925'  and  62920  are  the  r°°tS' 

QUESTIONS   FOR  EXERCISE. 

(1)  Required  six  numbers  whose  sum  and  product  shall  be  equal. 

Ans.  1,  2,  3,  4,  5,  and  -^-. 

(2)  Required  five  square  numbers  whose  sum  shall  be  a  square 

Ans.  1,  4,  9,  16,  and  j. 

(3)  Divide  the  number  3  into  four  rational  squares. 

16    1      9  49 

Ans.  — ,  — ,  — ,  and  — . 

2o   2o    2o  25 

(4)  Divide  unity  into  three  rational  squares. 

9      4  36 

AnS-  49'  49'  aQd  49" 

(5)  Find  two  numbers  whose  sum  is  a  cube,  and  difference  a  square. 

Ans.  1512  and  216. 

(6)  Find  two  numbers  whose  product  plus  their  sum  or  difference  is  eacb 
i  square. 

5        ,     5 
i  Ans.  —  and  4—. 

(7)  To  find  two  numbers, such,  that  when  each  is  multiplied  into  the  cubo 
of  the  other, the  products  will  be  squares. 

Ans.  2  and  8. 

(8)  To  find  two  square  numbers  whose  difference  is  40. 

Ans.  49  and  9. 

(9)  To  find  two  square  numbers,  such,  that  their  sum  added  to  their  prod- 
uct may  be  a  square  number. 

A  X         A4 

AnS-  9  and  9* 

(10)  It  is  required  toJind  two  whole  numbers,  such,  that  their  difference, 
the  difference  of  their  squares,  and  the  difference  of  their  cubes  shall  be  squares. 

Ans.  10  and  6. 

(11)  Find  two  numbers,  such,  that  the  sum  of  their  squares  shall  be  both 
a  square  and  a  cube. 

Ans.  75  and  100. 

(12)  Find  two  numbers  whoso  sum  shall  be  a  cub»  but  their  product  and 
quotient  squares. 

Ans.  25  and  100. 


46d  ALGKBRA. 

(13)  It  is  required  to  find  three  integral  sqiaro  numbers  that  shall  be  to 
arithmetical  progression. 

Ans.  1,  25,  and 

(14)  To  find  three  square  integral  numbers  in  har.monical  progression. 

Aits.  1225,  49,  and  2  >• 

(15)  To  find  three  numbers, such,  that  if  to  the  square  of  eacli  of  them  the 

sum  of  the  other  two  be  added,  the  three  sums  shall  be  all  squat 

8        ,  1G 
Ans.  1,  -,  andy. 

(1G)  It  is  required  to  find  three  whole  numbers,8iich,  that  if  to  the  square  of 
each  of  them  the  product  of  the  other  two  be  added,  the  sums  shall  be  squares. 

Ans.  9,  73,  and  3 

(17)  It  is  required  to  find  three  whole  numbers  in  geometrical  progression, 
such,  that  the  difference  of  every  two  of  them  shall  be  a  square  number. 

Ans.  5G7,  1008,  and  17 

(18)  It  is  required  to  find  three  integral  square  numbers, such,  that  the  dif- 
ference between  every  two  of  them  and  the  third  shall  be  a  square  number. 

Ans.  149s,  241s,  and  21 

(19)  To  fiud   three   square   numbers,  such,  that  the  sum  of  their  squares 
shall  also  be  a  square  number. 

,  144 
Ans.  9,  16,  and  -—jr. 

KmtJ 

(20)  To  find  throe  biquadrate  numbers  the  sum  of  which  shall  be  a  square. 

Ans.  12*,  154,  and  204. 
For  generalization  of  Diophantine  problems  in  certain  cases,  see  Bonny- 
castle's  Algebra.     See,' also,  Theory  of  Numbers. 


THEORY  OF  NUMBERS. 

i 

387.  We  have  already  had  occasion  to  demonstrate  some  propositions  which 

fall  under  this  head,  and  which  would  have  been  reserved  for  this  place  had 
they  not  been  required  for  the  elucidation  of  previous  parts  of  the  work. 

We  recur  to  one  or  two  of  these  for  the  purpose  of  exhibiting  some  of  the 
other  methods  by  which  they  may  bo  established. 

T.  To  prove  that  axb  =  bxa-     Suppose  d^>b  and  c  their  dillerence; 

.-.  aXt  =  (H c)b  =  l-  +  cb; 
i.  e.,  b  taken  b  times  and  c  taken  b  times,  and 

bxa  =  b(b+c)  =  b"+bc; 
■    ' .,  b  taken  b  times  and  also  c  times. 

We  perceive  that  the  product  <ixl>  will  be  the  Bame  ;ls  &X<*>  it"  the  partial 
product  rxb  is  equal  to  /'X'--  But,  by  similar  reasoning,  the  equality  of 
and  br  will  be  proved  by  the  equality  of  two  smaller  products,  <-,/  and  de;  and 
continuing  thus,  we  arrive  necessarily  at  the  case  where  the  two  factors  are 
equal,  or  at  the  case  where  one  of  them  is  equal  to  unity.  Tn  tho  first  i 
the  equality  is  manifest ;  in  the  Becond,  it  will  follow,  from  the  fact  that  /ixl 
is  //  as  well  as  1  x''-  Then  the  product  a  X  b  18  always  equal  to  the  product 
OXi- 


THEORY  OF  NUMBERS  4(j9 

IT.  To  demonstrate  that  NxaXi=Nx«i,  I  observe,  first,  that  the  prod- 
act  ah  is  nothing  else  than  a+a  +  a  +  >  &*•> tuo  number  of  these  terms  beingo. 
ThenNxai=Na  +  Na  +  ]Na  +  ,  <5cc,  to  £  terms,  =Na  X  b.         Q.  E.  D. 

III.  Nab=Nba  ;  for  Na=N+N+N-|-  ...  to  a  terms  ;  then,  to  multiply 
N«  by  b,  it  is  necessary  to  take  each  of  the  terms  b  times, 
thus  Na6=No+N6-fN&  .  .  .  =N6a.  Q.  E.  D. 

Corollary  1. — If  all  the  factors  of  N  be  1,  then  1  Xab  =  l  X  bu,  or  ab=ba. 
according  to  I. 

Corollary  2. — The  above  reasoning  applies  only  to  entire  factors.  The  prin- 
ciple is  equally  true,  however,  when  some  of  the  factors  are  fractions  ;  because, 
if  the  entire  factors,  which  are  combined  with  the  fractional  ones,  lie  written 
in  a  fractional  form  by  placing  unity  under  them,  all  the  factors  to  be  multi- 
plied together  will  be  fractions ;  the  product  of  these,  we  know,  is  obtained  by 
taking  the  product  of  the  numerators  and  denominators  separately,  which  are 
entire  numbers,  and  therefore  the  order  is  immaterial,  from  what  1ms  been 
proved  above. 

Corollary  3. — If  the  factors  be  incommensurable,  it  is  to  be  observed  that 
the  product  of  two  incommensurable  quantities  has  no  precise  meaning. 

P>ut  by  regarding  the  incommensurables  as  limits  to  which  approximating 
commensurables  tend,  since  the  above  reasoning  applies  to  the  latter,  and  their 
order  is  immaterial,  we  may  infer  that  the  order  is  immaterial  also  in  a  prod- 
uct of  incommensurable  factors. 

Corollary  4. — We  have  seen  that,  from  the  above  proposition,  it  follows  that 
the  order  of  factors  in  a  product  is  immaterial ;  hence  it  follows  that  if  a 
number,  P,  contains  the  factors  a,  b,  c,  &c,  it  is  divisible  by  their  product. 

Corollary  5. — If  a  number,  P,  is  divisible  by  another,  Q=a6c,  then  is  P 
divisible  by  each  of  the  factors  a,  b,  c. 

THE  FORMS  AND  RELATIONS  OF  INTEGRAL  NUMBERS,  AND  OF  THEIR 
SUMS,  DIFFERENCES,  AND  PRODUCTS. 

388.  I.  The  sum  or  difference  of  any  two  even  numbers  is  au  even  num- 
ber.    For,  let  A=2n  and  B=2n'  be  any  two  even  numbers  ;  then 

A±B=2»±2»'p2(»±»')=2n"' 

which,  being  of  the  form  2«,  is  an  even  number. 

II.  The  sum  or  difference  of  two  odd  numbers  is  even,  but  the  sum  of  three 
odd  numbers  is  odd. 

Let  A=2m-(-1,  B=2«'+l,  and  C=2n"-\-l,  be  three  odd  numbers;  then 
A  +  B  =  2m  +  2»'+  2  =  2»", 
and  A+B  +  C=2»+2w'+2«"+ 3=2»'"-f  1 ; 

the  former  having  the  form  of  an  even,  and  the  latter  of  an  odd  number. 

In  a  similar  way  it  may  be  shown, 

(1)  That  the  sum  of  any  number  of  even  numbers  is  even. 

'2)  That  any  even  number  of  odd  numbers  is  even,  but  that  any  odd  num- 
jer  of  odd  numbers  is  an  odd  number. 

(3)  "That  the  sum  of  an  even  and  odd  number  is  an  odd  number. 

(4)  That  the  product  of  any  number  of  factors,  one  of  which  is  even,  will 
be  an  even  number,  but  the  product  of  any  number  of  odd  nun  bers  is  odd 
and  hence,  again, 


470  ALGEBRA. 

(5)  Every  power  of  an  even  number  is  even,  and  every  power  of  an  odd 
number  is  an  odd  number. 

(6)  Henco  the  sum  and  difference  of  any  power  and  its  root  is  an  even 
number. 

For  the  power  and  root  will  be  either  both  even  or  both  odd,  and  the  sum 
or  difference  in  either  case  is  an  even  number. 

III.  If  an  odd  number  divide  an  even  number,  it  will  also  divide  the  half 
of  it. 

Let  A  =  2n,  B=2rc'-{-l  be  any  even  and  odd  number,  such  that  B  a  a 
divisor  of  A  ;  let  the  division  be  made,  and  call  the  quotient  p  ;  then  we  have 

,  2n=p(2n'+l); 

consequently  (4),  p  is  even,  or  of  the  form  2n"  ; 
hence  2n=2»"(2»'-|-l), 

and  2-^+l=n"; 

that  is,  n=|A  is  divisible  by  B,  if  A  itself  be  so. 

DEFINITIONS. 

389.  (1)  A  perfect  number  is  that  which  is  equal  to  the  sum  of  all  its  ah 

quot  parts,  or  of  all  its  divisors. 

6      6      6 
Thus,  6=--\---\--,  and  is,  therefore,  a  perfect  number. 

(2)  Amicable  numbers  are  those  pairs  of  numbers  each  of  which  is  equal  tu 
all  the  aliquot  parts  of  the  other.  Thus,  284  and  220  are  a  pair  of  amicable 
numbers,  for  it  will  be  found  that  all  the  aliquot  parts  of  284  are  equal  to  220, 
and  all  the  aliquot  parts  of  220  are  equal  to  284. 

(3)  Figurate  numbers  are  all  thoso  that  fall  under  the  general  expression 

w(n+l)(w-f  2)(«-f  3). . .  .(n+?n) 
1.2. 3. 4. ...(?/i+l)  ' 

and  they  are  said  to  be  of  the  1°,  2°,  3°,  &c,  order,  according  as  ?n  =  l 
2,  3,  &c 

(4)  Polygonal  numbers  are  the  si^is  of  different  and  independent  arith- 
metical series,  and  are  termed  lineal  or  natural,  triangular,  quadrangular 
or  square,  pentagonal,  &c,  according  to  the  series  from  which  they  are 
generated. 

(5)  Natural  numbers  are  formed  from  a  series  of  units  ;  thus  : 

Units,  1,  1,  1,  1,  1,  &c. 

Natural  numbers,  1,  2,  3,  4,  5,  &c. 

(6)  Triangular  numbers  are  tho  successive  sums  of  an  arithmetical  series, 
beginning  with  unity,  the  common  difference  of  which  is  1  ;  thus  : 

Arithmetical  spi  1,  2,  3,    4,     5,   &c 

Triangular  numbers,  1,  3,  6,  10,  1">.  6 

(7)  Quadrangular  or  square  numbers  are  the  sums  of  an  arithmetical  se 

ries,  beginning  with  unity,  and  the  common  difference  of  which  is  2  ;  thus  : 

Arithmetical  series,        1,  3,  5,    7,     9,    11,  &c. 

Quadrangular      or  > 


square  numbers, 


1.    I.  9,  16,  25,  36,  &c. 


THEORY  OF  NUMBERS.  471 

(8)  Pentagonal  numbers  are  the  sums  of  an  arithmetical  series,  beginning 
with  unity,  the  common  difference  of  which  is  3  ;  thus  : 
Arithmetical  series,         1,  4,    7,    10,  13,  16,  &c. 
Pentagonal  numbers,       1,  5,  12,  22,  35,  51,  &c. 

And,  universally,  the  m — gonal  series  of  numbers  is  formed  from  the  suc- 
cessive sums  of  an  arithmetical  progression,  beginning  with  unity,  the  com- 
mon difference  of  which  is  m — 2. 

DIVISIBILITY  OF  NUMBERS. 

390.  I.  The  product  of  two  numbers,  a  and  b,  is  divisible  by  every  number 
which  exactly  divides  one  of  the  two  facAxyrs  a  and  b. 

For  let  6  be  a  number  which  divides  b,  so  that  b=c6,  we  have  by  the  fore 
going  ab=acX6-     Then  ab,  divided  by  0,  gives  the  exact  quotient  ac. 

Corollary. — To  divide  a  product  of  several  factors,  divide  one  of  the  factors 
and  multiply  the  quotient  by  the  others. 

On  this  subject  we  must  observe  that  a  number  may  sometimes  divide  a 
product  when  it  will  not  divide  any  factor.  Thus,  20  divides  neither  12  nor 
15,  but  does  their  product,  180.  This  is  because  20  is  composed  of  factors 
some  of  which  are  found  in  12  and  others  in  15.  But  if  the  number  20  had 
no  common  factor  with  one  of  the  factors,  it  must  divide  the  other.  (See 
Art.  84,  note.) 

II  If  there  be  n  numbers,  each  of  them  divisible  by  k,  then  is  their  product 
divisible  by  kn. 

For  a=kq,  b=.kq',  c=kq".  .  .     .-.     abc...=kn.w, 

w  being  equal  to  q  X  q'  X  q"  X 

III.  The  sum  of  several  numbers,  a+b-j-c-j-d.  is  divisible  by  a  number,  k, 
when  the  sum  of  th&  remainders  obtained  by  dividing  each  by  k  is  divisible  by 
this  number. 

For  a=kq-\-r,  b=kq'-\-r',  c=kq"-\-r",  &c. 

...  a  +  b  +  c+ d=k{q+q'+q"  +  ,  &c.)-f-r+r/+r"+,  &c. 
Whence  it  is  evident  that  a-\-b-\-c,  <kc,  is  divisible  by  k  when  r-f-r'4-r", 
&c,  is. 

IV.  The  difference  of  two  numbers,  a  and  b,  is  divisible  by  a  number,  k, 
when,  if  each  be  divided  by  k,  the  remainders  are  equal. 

For  a=kq-\-r,  and  b=kq'-\-r 

.'.  a  —  b=k(q — q'). 

V.  Every  number  consisting  of  units,  tens,  hundreds,  Sfc,  is  divisible  by  a 
number,  k,  when  the  sum  of  the  jiroducts  of  the  number  of  units,  tens,  8fc,  by 
the  remainder,  after  dividing  the  units,  tens,  Sfc,  each  by  k,  is  divisible  by  this 
number. 

For,  representing  by  A,  B,  C,  &c,  the  quotients,  and  by  a,  )3,  y,  &c.,  the 
remainders  of  the  units,  tens,  &c,  by  k,  we  have 

10n     =Afc-f  a  .-.         a  .  10"     =aA.k  +  aa 

10n-'=BA"+/3  b  .  10f-'=  bBk+bp 

10n-2  =  C&+y  c  .  10°--*=  cCk+cy 


103    =Dfc-f  <J                      d  .  10*    =dDk+dd 
101     =EJfc-j-e                        e  .  101     =eEk  +  ee 
10°    =  ...  1  /.  10°    = / 


472  ALGEB1LA. 

VI.  The  proauct,  P,  of  several  numbers,  a,  b,  c,  d,  .  .  .  is  divisible  by  a 
number,  k,  only  when  the  product  of  the  remainders,  after  dividing  each  of  the 
factors  by  k,  is  so  divisible. 

For,  let  a=kq-\-a,  b  =  k  /-{-  I,  c=k/'-{-y,  &c, 

.-.  ab=kz-\-a.(3. 
abc=kz-\-ap.y,  &c. 

VII.  The  product,  P,  of  several  factors,  a,  b,  c,  d,  .  .  .  is  divisible  by  a  prime 
number,  k',  only  when  one  of  the  factors  is  divisible  by  Otis  prime  n  umber 

For,  let  a=k'q-\-a,  b=k'q'-\-p,  c=zk'q"-\-y,  &c, 

.'.  V  =  k'z-\-a.[i.y  .  .  . 

Therefore,  if  k'  divido  P,  it  must  divide  a,  3,  y  .  .  . 

But  k'  is  not  fouud  among  the  factors  a,  (3,  y,  .  .  .  since,  being  remainders 
to  the  divisor  k',  they  are  all  less  than  it.  Neither  is  k'  any  combination  of 
them,  since  it  is  supposed  to  be  a  prime  number.  Hence  a,  j3,  y,  .  .  .  and 
therefore  P  is  divisible  by  k'  only  when  one  of  the  remainders  =0. 

VIII.  If  the  factors,  a,  b,  c,  .  .  .  of  a  product,  P,  are  prime  to  k,  then  is  the 
product  not  divisible  by  k. 

For,  if  k  bo  an  absolute  prime  number,  this  follows  from  VII.     Again,  if 

k  be  a  multiple  of  a  prime  number,  as  p'v;  then,  if  P  be  divisible  by  k,  \\t 

have 

P      a.b.c.... 

~r= ; =m  ••.  a.b.c...  =mp'v ; 

k  p' .  v  1 

thereforo  a.b.c...  must  be  divisible  by  p' ,  which  by  VII.  is  impossible. 

391.  I.  Problem. — To  find  all  the  divisors  of  any  number  ivkatever.  The 
first  thought  which  presents  itself  is  to  try  successively  as  divisors  each  of  the 
numbers  1,  2,  3,  &c.,  to  N.  But  this  groping  process  may  be  abridged.  Lei 
D  be  a  divisor  of  N,  and  D'  the  quotient,  we  have  DD'  =  N,  or,  under  anoth- 
er form,  DD'=  -/N  X  VN  ;  then,  if  D_is  <  y/N,  D'  will  be  >  -/N.  Tb 
after  having  found  all  the  divisors  <  -\/  S,  the  quotients  which  shall  have  been 
obtained  in  dividing  N  by  theso  divisors  will  be  the  divisors  >  -J  V 

For  example,  lot  N=360.  The  square  root  of  360  is  comprised  between 
18  and  19 ;  thus,  we  divido  3G0  only  by  the  numbers  1,  2,  3  ...  18.  [a  this 
manner  we  find  all  the  divisors  of  3G0,  to  wit : 

1,       2,       3,      4,     5,     6,     8,     9,    10,   12,  15,   18. 
360,   180,  120,  90,  72,  60,  45,  40,  36,  30,  24,  20. 

392.  II.  Problem. — To  form  a  table  of  prime  numbers.     When  the  ;• 
proceeding  produces  no  divisor,  the  number  ia  a  prime  number.     To 
the  long  calculations  necessary  in  these  cases,  tallies  have  been  construt 
whirl i  co  itain  the  prime  numbers  up  to  certain  limit 

The   in"  !    simple  manner  of  constructing  it  is  to  write  i  -ion  the 

series  of  an  even  numbers  3,  5,  7,  &C.,  to  such  n  limit  as  we  seek,  and  to  efl 
all  the  multiples  of  3,  of  5,  of  7,  evrc.     It  is  evident  that  the  prime  numbers 
are  all  that  remain.      At  the  head  of  these  numbers  it  must  not  be  forgotten  to 
place  1  and  2. 

Nothing  is  easier  than  to  know  what  multiples  to  efface.     Those  of  3  are 

*  The  student  is  referred  to  the  tables  of  Burckharlt,  ia  which  the  prinii-  Mjnben  ex 
tend  to  .103G000. 


THEORY  OF  NUMBERS.  473 

tound  by  counting  the  numbers  3,  5,  7,  &c,  in  threes,  setting  out  from  5 ; 
those  of  5  in  counting  them  in  fives,  beginning  with  7,  and  so  on.  ■ 

393.  Remark  I. — The  series  of  prime  numbers  is  unlimited.  For,  suppose 
it  to  be  otherwise,  and  that  n  is  the  greatest :  if  we  form  the  product 
P=2.3.5  .  .  .  n,  which  contains  all  the  prime  numbers,  then  P  +  l,  which 
>n,  must  be  divisible  by  some  one  of  these  numbers  ;  but  this  is  impossible, 
because  there  will  always  be  the  remainder  1.  Then  it  is  impossible  that  the 
series  of  prime  numbers  should  be  limited. 

II.  In  comparing  all  numbers  with  multiples  of  the  same  number,  we  are 
led  to  present  them  under  different  forms,  of  which  use  is  often  made.     For 
example,  if  wo  compare  them  with  multiples  of  6,  they  may  be  represented 
first,  by  one  of  the  six  formulas, 

6.r,  6x+l,  G.r+2,  Gx+3,  G.r+4,  6x-f  5, 
in  which  x  is  any  whole  number  whatever. 

But  if  we  wish  to  consider  only  prime  numbers,  it  is  necessary  to  preserve 
only  the  two  formulas, 

6z+l  and  Gx-{-h  ; 

because  the  others  give  numbers  divisible  by  2  or  by  3. 

We  can  also,  in  place  of  6.r+5,  write  6(.r+l)  — 1  or  G.v— 1,  since  x  is  any 
entire  number  whatever.  Thus  all  the  prime  numbers  except  2  and  3,  which 
are  divisors  of  6,  are  comprised  in  the  formula 

N  =  G.r±l. 
The  reasoning  woidd  be  analogous  for  any  other  number  than  6. 

394.  III.  Problem. —  To  decompose  a  number  into  prime  factors,  and  to  find 
afterward  all  its  divisors. 

A  number  N,  if  it  be  not  a  prime  number,  can  be  represented  by  the  product 
of  several  prime  numbers  a,  b,  c,  &c,  raised  each  to  a  certain  power,  60  that 
we  can  always  suppose  N=am6ncP  .  .  .  This  is  the  decomposition  which  it  is 
required  to  effect. 

Take,  for  example,  the  number  504.  Divide  it  first  by  2  as  many  times  as 
possible  ;  we  find  thus, 

504=252  X  2=126  X  2  X  2  =  63  X  2  X  2  X  2. 
Then  divide  63  as  many  times  as  possible  by  3,  which  is  the  sirallest  prime 
number  greater  than  2  : 

63  =  21X3=7X3X3. 
Then  we  have 

504  =  7X3X3X2X2X2, 

or,  rather,  under  another  form, 

504=23X3:X7. 

The  divisions  by  3  have  led  to  the  quotient  7.  If  the  quotient  had  not  been 
a  prime  number,  we  should  have  continued  the  operations  by  trying  success- 
ively the  other  prime  numbers,  5,  7,  &c. 

We  can  now  readily  form  all  the  divisors  of  504.  They  are,  in  fact,  the 
numbers  which  we  obtain  in  taking  all  the  prime  factors  one  by  one,  two  by 

*  Conceive  a  board  pierced  with  holes  in  -which  the  numbers  3,  5,  7,  &c.,  are  placed  in 
order.  Then,  as  we  arrive,  in  counting  them  by  threes,  fives,  lVc,  at  the  multiples  to  be 
effaced,  suppose  these  multiples  to  fall  tln-ough  the  holes,  there  will  remain  only  prime 
numbers.     Such  was  the  famous  sieve  of  Eratosthenes,  of  Alexandria,  who  lived  280  B.C 


47  4  ALGEBRA. 


two,  Arc.     That  we  may  be  sure  not  to  omit  any  :  ivisor,  we  adopt  the  fol- 
lowing arrangement : 

1, 

504     2      2, 
252     2      4, 
126     2      8, 
63     3      3,      6,     12,    24, 
21     3      9,     18,    36,    72, 
7     7      7,     14,    28,    56,     21,     42, 
84,  168,  63,  126,  252,  504. 
The  first  column  on  the  left  contains  the  given  number  and  the  quotient  ol 
the  successive  divisions.     By  the  side  of  these  numbers,  in  a  second  column, 
are  written  the  prime  numbers,  which  we  employ  as  divisors,  and  which 
are  the  prime  factors  of  the  number  504.     Finally,  we  place  at  the  right  of 
this  column  all  the  divisors  of  504 ;  and  I  now  proceed  to  state  how  we  obtain 
them. 

At  the  top  of  the  third  column,  but  on  the  line  above,  that  which  contains 
504,  we  write  unity,  which  may  be  regarded  as  the  first  divisor  of  504.  We 
multiply  this  unity  by  the  first  number  of  the  second  column,  and  thus  obtain 
the  divisor  2,  which  we  write  by  the  side  of  this  first  prime  number.  We 
next  multiply  1  and  2,  the  divisors  already  found,  by  the  second  number  of  the 
second  column,  and,  neglecting  the  product  1X2,  or  2,  which  has  already  been 
found,  we  obtain  the  new  divisor  4,  which  is  written  on  a  line  with  the  last 
multiplier.  We  proceed  in  the  same  manner,  multiplying  the  number  of  the 
second  column  on  the  horizontal  line  which  wo  are  forming  by  each  of  the 
numbers  above  it  in  the  third  column  successively,  until  we  multiply,  finally. 
by  the  last  number  of  the  second  column,  which  gives  a  last  series  of  divisor-. 
which  series  will  always  be  terminated  by  the  given  number. 

When  we  know  the  prime  factors  of  a  number,  we  can  find  its  divisors  by 
another  process.  Suppose  that  a  number  N,  when  decomposed  into  prime 
factors,  gives 

N=ami"cP  .  .  .; 
the  divisors  of  N  will  be  represented  by  the  formula  am'ba'ci"  . .  ..  in  which  tin 
exponents  to',  n',  p* . . .  can  not  surpass  to,  n,  p . . . 

Hence  we  know  that  these  divisors  will  bo  the  different  terms  which  w« 
obtain  in  effecting  the  product 

P  =  (l  +  «+«N am)(l  +  /,-fZ,:+  . . .  i'>)(i-f_(.+c-_j_  .  .  .  cp) 

395.  Remarks. — Tho  multiplication  of  tho  first  two  polynomes  gives  a 
number  of  terms  equal  to  (m-fi)(«-|_i) ;  consequently,  thai  of  the  first  three 
polynomes  gives  a  number  equal  to  (m-r"l)(n+l)(jp-f-l),  and  so  on;  hence, 
the  number  of  all  the  divisors  of  N  is  sed  by  the  formula 

(m+l){n+l)(p+l) 

We  also  see  that  P  is  tho  sum  of  all  theso  divisors.     But  we  know  that  the 

<;"■+'  — 1 


polynomes  which  composo  P  are  respectively  equal  (Art  23)  to 

</  —  1 
6"+'  — l 
-f)_l  »  &c. ;  hence,  tho  sum  of  all  the  divisors  of  N  can  bo  expressed  by  the 

formula 


THEORY  OF  NUMBERS.  475 

aH-l_l        fcn+l  —  1        C?+l  —  1 


a — 1  o  —  1  c  —  1 


For  example,  taking  N=504=23X32X?,  we  shall  have  to=3,  ra=2, 
psssl.  Hence  the  number  of  divisors  of  504  will  be  4x3X2=24,  and  the 
sum  of  all  the  divisors  will  be 

2<— 1      33— 1      73— 1 

IT^l  X  3^1  X  731  = 15  X  13  X  8  =  156°- 

396.  IV.  Problem. — How  many  times  is  a  prime  number,  0,  factor  in  a 
series  of  natural  numbers,  from  1  to  n  ?  or,  in  other  words,  what  is  the  highest 
power  of  0  which  divides  the  product  1 . 2 . 3  . . .  n  ? 

Let  n'  be  the  entire  part  of  the  quotient  of  n  by  0.  In  tli9  proposed  series 
of  natural  numbers  we  find  the  n'  factors,  0,  20,  30 . .  . .,  of  the  product 
6.26.30. .  .n'Q;  and  it  is  clear  that  they  are  the  only  numbers  of  the  series 
which  are  divisible  by  6.     This  product  can  be  written  thus : 

1 .  2  .  3  ...  n'  X  0n'. 
Hence  we  shall  obtain  the  required  power  of  6  by  multiplying  0n/  by  the  high- 
est power  of  0,  contained  in  the  product  1 . 2 . 3  . . .  n'. 

The  same  reasoning  may  bo  repeated  with  reference  to  this  product ; 
hence,  calling  n"  the  entire  part  of  the  quotient  of  n'  by  0,  we  readily  perceive 
that,  the  highest  power  of  0  contained  in  the  last  of  the  above  products  is  com- 
posed of  the  power  0n"  multiplied  by  the  highest  power  of  0  which  is  contain- 
ed in  the  series  1.2.3  .  .  .  n". 

In  like  manner,  calling  n'"  the  entire  part  of  the  quotient  of  n"  by  6,  we  are 
led  to  seek  the  highest  power  of  0  contained  in  the  product  1.2.3  .  .  .  n'". 

We  continue  this  process  till  we  arrive  at  a  quotient  <C,0.  For  the  sake  of 
definiteness,  suppose  that  n'"  is  this  quotient ;  then  we  conclude  that  the 
highest  power  of  0  contained  in  the  given  product  1 . 2 . 3  .  . .  n  is  6a,+a"+n'". 

For  example,  suppose  we  wish  to  l?how  what  is  the  highest  power  of  7 
which  divides  the  product  1.2.3  .  .  .  1000. 

We  make  n=1000,  and  taking  only  tltt   entire  parts  of  the  quotients,  we 

shall  have 

1000  142  20 

-—-  =  142,  —=20,  — =2. 

7  7  7 

The  sum  of  these  quotients  being  164,  it  follows  that  the  required  power  is  716*. 

397.  Corollary. — Let  m,  n,  p,  q  be  entire  numbers,  such  that  we  have 
m=n-\-jy-\-q-\-  .  .  .  ;  the  expression 

1.2. 3. 4. to 
1.2 » X  1  •  2 pXl-2 qX,  &c. ^ 

will  always  represent  an  entire  number.    To  prove  this,  let  0  be  a  prime  factor 
of  the  denominator  ;  wo  shall  have 

to     n      p      q 

Calling  these  entire  quotients  to',  n',  p',  q'  .  .  .  .,  we  shall  have  also 

to'=  or  >«'+_Z''+9'+i  &c. 
If  wo  divide  again  by  0,  and  call  the  new  entire  quotients  m",  n"  .  .  .  .,  we 
shall,  in  like  manner,  have 

to"=  or  >n"+p"+q"+ ,  &c. 
We  continue  this  process  as  long  as  the  quotients  are  not  all  less  than  6 


476  ALGEBRA. 

Then  adding,  we  shall  have 

'+„*"+...)=  or  >(n'+n"+...)  +  (jp'+p"+...)+(?'+5r"+-)+.  *c. 

l!iii  these  different  sums  make  known  the  highest  powers  of  0,  by  which  we 

divide  the  products  which  compose  expression  (1);   hence  there  is  no 

ie  factor  in  the  denominator  which  does  not  exist  of  a  power  at  least  equal 

in  the  numerator  of  the  fraction.     This  expression,  therefore,  represents  an 

entire  number. 

393.   Perfect  numbers  are  expressed  or  determined  as  follows  : 

Find  2" —  1,  a  prime  number,  then  will  N=2B— I(2" —  1)  be  a  perfect  number. 

For,  from  what  has  been  demonstrated  in  the  preceding  section,  the  sum  of 

2" l      (2° 1): 1 

all  the  divisors  of  this  formula  will  bo  represented  by  — — —  x  77^ — rr — r ; 

because  2"  —  1  is  a  prime  by  hypothesis.     But  in  this  expression  1  is  inclu 
as  a  divisor,  which  must  be  excluded  in  the  case  of  perfect  numbers  :  pxcIi 
sive  of  this,  therefore,  the  formula  will  be 
On_!      (2n_  l)s_l 

X: '. On-2 1  /->n T  \ 
(On ]\ I          ~                 L\~           *■> 

(2»  —  l)x(2n  — 1  +  1)  —  2n-'(2n  —  1)  = 
2(2n— l)2n+1  —  J"   '(-'"  —  l)=2n-1(2"  — 1)  =  N, 
that  is,  the  sum  of  all  the  aliquot  parts  of  N,  exclusive  of  itself,  or  of  1  as  ;i 
divisor,  is  equal  to  N,  and  is,  therefore,  by  the  definition  a  perfect  number. 
The  only  perfect  numbers  known  are  tho  following  eight: 

G,  3355033G, 
28,  8589869056, 
496,  137438691328, 
8128,  23058430081399521  ■-■-. 

399.  To  find  a  pair  of  amicable  numbers  N  and  M,  or  such  a  pair  that  each 
shall  be  respectively  equal  to  all  tho  divisors  of  the  other. 

Make  N=amincP,  &C,  and  M=af,Pvyn ;  then,  according  to  the  definition  and 
from  what  has  been  demonstrated  in  the  last  section,  we  must  have 
am-H_l      &n+i_i      CP+1  —  1 

a  —  1  0  —  1  c — 1  ' 

ae+i_l      pH-i-l      y*+i_l 

T=rxi^rx^=T-=?'r+N- 

Find,  therofore,  such  a  power  of  2,  as  2r,  that 

3.2r— 1,  6.2r— 1,  and  18.  2r  —  1 
may  bo  all  prime  numbers  ;  then  will 

N=2r+1(iaud  M  =  2r+'6c 

be  the  pair  of  amicable  numbers  sought. 

The  least  three  pair  of  amicable  numbers  are 

284,  220. 

17296,  18416, 

9363583,  9437056. 

400.  We  shall  hero  introduce  the  student  to  tho  nomenclature  and  notation 
of  Gauss,  given  in  his  Disquisitiones  Arithmetics,  which  is  now  generally 
adopted  by  writers  upon  the  theory  of  numbers. 


CONGRUENCE  OF  NUMBERS.  477 

CONGRUOUS  NUMBERS  IN  GENERAL. 

101.  If  a  number  a  divide  the  difference  of  the  numbers  b  and  c.  b  and  c  are 
said  to  be  congruous  with  reference  to  a  ;  if  not,  incongruous.  The  quantity 
a  is  called  the  modulus  ;  each  of  the  numbers  b  and  c  a  residue  of  the  other  in 
the  first  case,  a  non-residue  in  the  second. 

The  numbers  may  be  either  positive  or  negative,  but  entire.  As  to  the 
modulus,  it  ought  evidently  to  be  taken  without  regard  to  the  sign. 

Thus,  — 0  and  -{-16  are  congruous  with  reference  to  the  modulus  5;  — 7 
is  a  residue  of  15  with  reference  to  the  modulus  11,  and  not  a  residue  with 
reference  to  the  modulus  3. 

Zero  being  divisible  by  all  numbers,  every  number  may  be  regarded  as  con- 
gruous with  itself  with  reference  to  any  modulus  whatever. 

All  the  residues  of  a  given  number,  a,  with  reference  to  a  given  number,  m, 
are  comprised  in  the  formula  a -{-km,  I:  being  an  entire  indeterminate  num- 
ber.    This  is  self-evident. 

The  congruence  of  two  numbers  is  expressed  by  the  sign  EE,  joining  to  it 
the  modulus,  when  necessary,  in  a  parenthesis,  thus  :* 

—  16  =  9(mod.  5),  — 7=15(mod.  11). 

402.  Theorem. — Let  there  be  m  entire  successive  numbers,  a,  a-f-1,  a-J-2, 
...a-f-m  —  1,  and  another,  A;  one  of  the  former  will  be  congruous  with  A, 
with  reference  to  the  modulus  m,  and  but  one. 

a — A 

For  if is  entiro,  a  =  A;  if  it  is  fractional,  let  k  be  the  nearest  entire 

m 

n  a — A- 
number;  above,  if be  positive;  below,  if  it  be  negative;  A-\-km  will 

fall  between  a  and  a-\-m,\  and  will  bo  the  number  sought;  but  it  is  evident 

a — A  a-{-l — A 

that  the  quotients  ,  ,  &c,  are  comprised  between  k — 1  and 

1  m  m  r 

A"-f-l)t  therefore  one  of  them  only  can  be  entire. 

403.  It  follows  from  this  that  every  number  will  have  a  residue  as  well 
in  the  series  0,   1,  2...m  —  1,  as  in  the  series  0,   — 1,   — 2... — (m  —  1\ 
They  are  called  minima  residues ;  and  it  is  evident  that,  unless  zero  is  the 
residue,  there  will  be  two,  the  one  positive  and  the  other  negative.     If  they 

are  unequal,  the  one  will  be  <^-^\  if  they  are  equal,  each  of  them  =^-,  with- 
out regard  to  the  sign;  from  which  it  follows,  that  any  number  whatever  has 
a  residue  which  does  not  surpass  the  half  of  the  modulus;  this  is  called  the 
absolute  minimum  residue. 

For  example  :  — 13  relative  to  the  modulus  5,  has  for  a  positive  minimum  res- 
idue 2,  vWiich  is  at  the  same  time  its  absolute  minimum,  and  — 3  for  its  nega- 
tive minimum  residue;    -j-5,  with  reference  to  the  modulus  7,  is  itself  its 

*  The  analogy  between  equality  and  congruence  led  Legendre  to  employ  the  sign  of 
equality  itself.     This  modification  of  it  has  been  introduced  by  Gauss  to  avoid  ambiguity. 

t  This  maybe  seen  from  the  equality  —   —  =A — n,  where  ?j<w. 

?n 

i  This  may  be  seen  by  observing  that =  —  - — I —   and  it  is  not  dill  the   nunie- 

J  *  °  tin  in        in 

ratorof—  increases  to  m  that  the  quotient  Jc  increases  to  £-f-l. 
in 


•173  ALGEBRA.  s 

;■■  residue;   — 2  is  the  negativb  •ninimum  residue,  and,  at  the 

same  time,  the  absolute  mini  mum. 

404.  The  following  consequences  follow  from  tho  above  : 

Numbers  which  are  congruous  with  reference  to  a  composite  modulus  are  so 
with  reference  to  any  of  its  divisors. 

If  several  numbers  arc  congruous  with  the  same  number  with  reference  to  the 
same  modulus,  tkey  will  be  congruous  with  each  other  with  reference  to  this 
modulus. 

The  same  modulus  must  be  supposed  in  what  follows  : 

Congruous  numbers  havo  the  same  minima  residues  ;  incongruous  have 
different. 

405.  If  the  numbers  A,  B,  C,  &c. ;  a,  b,  c,  &c,  are  congruous  each  to  each, 
i-  e.,  AEEa,  BEE&,  &c,  we  shall  have 

•  A  +  B  +  C  .  .  .   =  «  +  i  +  c  .  .  . 
Tf  A~  a,  BEE  b,  we  have  also  A — BEE  a  —  b. 

406.  If  A  =.  a,  tee  have  also  k\  EE  ka.  * 

If  k  is  positive,  this  is  but  a  particular  case  of  the  preceding  article,  in 
which  A  =  B  =  C  .  .  .  and  a  =  b~c  .  .  . 

If  1c  is  negative,  —Jc  will  be  positive;  then  — A.-AEE  lea  .-.  kA  =  ka. 

If  A  =  a,  BEE  b,  then  ABEEai;  because  ABEE  AB EE  Ab  =  ba. 

107.  If  the  numbers  A,  B,  C  .  .  .  =  a,  b,  c  .  .  .,  each  to  each,  then 

ABC  .  .  .  =abc  .  .  . 
for,  by  the  preceding  article,  ABEEafc;    for  the  same  reason,  ABC  =  ab<, 
and  so  on. 

By  taking  all  the  terms,  A,  B,  C  .  .  .  equal,  and  a,  b,  c  .  .  .  also  equal,  if 
A=a,  A*EEa*. 

408.  Let  X  be  a  function  of  the  indeterminate  x  of  the  form 

A.ta-fB;r6+Crc+,  See., 
A,  B,  C  .  .  .  being  any  entire  numbers  whatever.     If  we  give  to  x  congruous 
values  with  reference  to  a  certain  modulus,  the  resulting  values  for  X  will  be 
congruous  also. 

Let/  and  g  be  congruous  values  of  x ;  by  the  preceding  articles, /"  I 
and  kf"~kg" ;  in  the  same  way  we  have  Bf''  =  Bgb,  &c. 

This  theorem  may  be  easily  oxtended  to  functions  of  several  indetermi 
nates. 

409.  If,  then,  we  substitute  in  place  of  r  all  entire  consecutive  numl 
and  seek  tho  minima  residues  of  tho  values  of  X,  they  will  form  a  series  in 
which,  after  an  interval  of  m  terms  (m  being  the  modulus),  the  same  terms 
will  be  again  presented;  that  is  to  say   this  Belies  will  be  formed  of  a  period 
of  in  terms  repeated  indefinitely. 

Let  there  be,  for  example,  X=r  —  -V-f-G,  and  ///  =  ">;  for  x=0»  1,  2,  •'?. 
&c. ;  the  values  of  X  give  for  positive  minima  residues  1,  4,  3,  4,  3,  1.  i.  &c, 
or  the  five,  1,  4,  3,  4,  3,  arc  repeated  indefinitely  ;  and  if  wq  continue  the 
series  in  the  contrary  direction,  that  is,  if  we  give  to  x  negative  values,  the 
same  period  will  reappear  in  an  inverse  onler:  whence  it  follows  that  the 
series  contains  no  other  terms  than  those  which  compose  the  period. 

410.  Then,  in  this  example,  X  can  nol  become  =  0,  nor  EE*J(nu>d.  5);  and 
still  less  =0  or  —'J';  from  which  it  follows  thai  the  equations  .;•'• — 8x4-6s0 
ami  r3— 8x-4-4=0  have  not  entire  roots,  and,  consequently,  not  rational  roots. 
V.  <•  Bee,  'u  general  that  when  X  is  of  the  form 


CONGRUENCE  OF  NUMBERS.  479 

Xn+Axn-»  +  B.r"-2+,  &c,  +  N, 

A.,  B,  C  .  . .  being  entire  quantities,  and  n  entire;  and  positive,  the  equation 
X=0  (a  form  to  which  every  algebraic  equation  may  be  reduced)  will  have  no 
rational  root,  if  it  happen  that,  for  a  certain  modulus,  the  congruence  X  =  0  be 
not  satisfied. 

411.  Many  arithmetic  theorems  may  be  demonstrated  by  tin;  aid  of  the 
foregoing  principles,  as,  for  instance,  the  rule  for  determining  whether  a  num- 
ber is  divisible  by  9,  11,  or  any  other  number. 

With  reference  to  the  modulus  9,  al!  the  powers  of  10  are  congruous  with 
unity;  then,  if  the  number  is  of  the  form  ir-\-lQb-\-100c-^-lQQ0d-\-,  &c,  it 
will  have,  with  reference  to  the  modulus  9,  the  same  minimum  residue  ^ 
a-\-b-\-c-\-,  &c.  It  is  clear  from  this,  that  if  we  add  the  figures  of  the  number 
without  regarding  their  place  value,  the  sum  obtained  and  the  proposed  num- 
ber will  have  the  same  minimum  residue.  It",  then,  this  last  is  divisible  by  9, 
the  sum  of  tho  figures  will  bo  also,  and  only  in  this  case.  It  is  the  same  with 
the  divisor  3. 

Many  of  the  properties  of  prime  numbers,  the  divisibility  of  products  already 
given,  &c,  may  be  demonstrated  by  the  aid  of  this  system,  but  we  shall  not 
repeat  them. 

412.  The  term  congruence  is  analogous  to  equation,  and  the  determination 
of  such  values,  for  an  indeterminate  x,  as  to  produce  congruence  in  expression, 
is  called  resolving  them.     There  are  congruences  resolvable  and  irresolvable. 

Congruences  are  also  divided,  like  equations,  into  algebraic  and  transcend- 
ental. Those  which  are  algebraic  are  divided,  again,  into  congruences  of  the 
first,  second,  and  higher  degrees.  There  are  congruences,  also,  containing 
different  unknown  quantities,  of  the  elimination  of  which  Gauss  treats. 

413.  The  congruence  oar-j-oEEc  may  be  solved  when  its  modulus  m  is 
prime  with  a  ;  thus,  let  e  bo  the  positive  minimum  residue  of  c  —  b.  We  find 
necessarily  a  value  of  x<0«,  such  that  the  minimum  residue  of  the  product 
ax,  with  reference  to  the  modulus  m,  shall  be  e.  Call  v  this  value,  and  we 
shall  have 

av  =  e  =  c — b ; 
then  av-{-b~EEc(mod.  m). 

Here  v  is  called  the  root  of  the  congruence.  It  is  evident  that  all  the  num- 
bers congruous  with  v,  with  reference  to  the  modulus  of  tho  congruence,  wil 
also  be  roots  (Art.  408).  It  is  »lso  evident  that  all  the  roots  should  be  con- 
gruous with  v;  in  fact,  if  t  be  another  root,  we  have  av-\-b  =  at-{-b ;  then 
at  =  av;  and  therefore  v=Et.  We  may  therefore  conclude  that  the  congru- 
ence xEE  t>(mod.  ?/i)  gives  the  complete  resolution  of  the  congruence  ax-\-b  EzEc. 

The  foregoing  exposition  will  serve  to  show  how  the  algorithm  of  (iauss 
connects  itself  with  the  indeterminate  analysis,  and  we  shall  hero  quit  the 
subject. 

414.  No  algebraical  formula  can  contain  prime  numbers  only. 

Let  2,  +  ax-{-TX'-^s^  &c-' 

represent  any  general  algebraical  formula.  It  is  to  be  demonstrated  that  such 
values. may  be  given  to  x,  that  the  formula  in  qui- 'ion  shall  not.  with  that  value 
produce  a  prime  number,  whatever  values  are  given  to  p.  q,  r,  cVc. 

For  suppose,  in  the  urst  place,  that  by  making  x=m,  the  formula 


480  ALGEB11A. 

V  =j>-\-qm-\-rm'l-\-sm:i-Jr.  &c, 
is  a  prime  number. 

And  if  now  we  assume  xr=»i-|-^P,  we  have 

P= P 

<F= 7m+?'?p 

rx-= r/zr  +  'J/vz/ol'-fro-P2 

ix'= s;/i:,+3s?/t,2^P  +  :j.s  m    P  +sfP3 

&c.  cVc. 

Or 

p -\- q.r -\-r.v- -\- sia =(p -\- qm -\- riir -\- siir -{- ,  &c.,)  + 
P w  _|_  2rm0  +  3s»t-<j>)  +  Ps(r^ + 3smf)  -f   .  I ' 
=  P  +  P(^+2m04-35m*f.)  + 
P-(7-^4-3sm^)+s£3P3. 
But  this  last  quantity  is  divisible  by  P ;  and,  consequently,  the  equal  quantity 

p-\-qx-{-rx'--\-sx*,  &c., 
is  also  divisible  by  P,  and  can  not,  therefore,  be  a  prime  number. 

Hence,  then,  it  appears,  that  in  any  algebraical  formula  such  a  value  may 
bo  given  to  the  indeterminate  quantity  as  will  render  it  divisible  by  some  other 
number  ;  and,  therefore,  no  algebraical  formula  can  be  found  that  contains 
prime  numbers  only. 

But, although  no  algebraical  formula  can  be  found  that  contains  prime  num- 
bers only,  there  are  several  remarkable  ones  that  contain  a  great  many;  thus, 
r3+.r-|-41,  by  making  successively  ar=0,  1,  2,  3,  4,  &c,  will  give  a  series 
41,  43,  47,  53,  Gl,  71,  &c,  the  first  forty  terms  of  which  are  prime  numbers 
The  above  formula  is  mentioned  by  Euler  in  the  Memoirs  of  Berlin  (1772, 
p.  36). 

To  the  above  we  may  add  the  following:  ,r:-|-r+]"  an^  2.r':  +  29  ;  the 
former  has  17  of  its  first  terms  prime,  and  the  latter  29. 

Fermat  asserted  that  the  formula  2,n-f-l  was  always  a  prime,  while  m  was 
taken  any  term  in  the   series   1,  2,   4,   8,   16,  &c.  ;    but   Euler  found   that 

4-1=641  X  6700417  was  not  a  prime. 

■11">.  If  a  and  6  bo  any  two  numbers  prime  to  each  oilier,  ami  each  of  the 
terms  of  the  series 

6,  26,  36,  46,  &c,  {a  — 1)6 
be  divided  by  a,  they  will  each  leave  a  different  remainder.  For  if  any  two 
of  these  terms,  when  divided  by  a,  leave  the  same  remainder,  let  them  be  rep 
ented  by  .r6,  yb  ;  then  it  is  obvious  thai  .*/' — yb  would  be  divisible  by  a,  or 
(r — 7)6  would  be  divisible  by  a.  But  this  is  impossible,  because  a  is  prime  to 
6,  and  x — y  is  less  than  a;  therefore  b{x —  >/)  is  not  divisible  by  a,  but  it 
would  be  so  divisible  if  the  terms  xb,  yb  left  the  same  remainder;  these  do 
not,  therefore,  leave  the  same  remainder;  consequently,  every  term  of  the 
series 

6,  26,  36,  Arc,  (a— 1)6, 

divided  by  a,  will  leave  a  different  remainder. 

DEDUCTIONS. 

416.  Since  the  remainders  arising  from  the  division  of  each  term  in  the  -• 

6,  26,  36,  &c,  (a— 1)6 
oy  a  are  different  fr)m  each  other,  and  <*  —  1  in  number,  and  each  of  thera 


CONGRUENCE  OF  NUMBERS.  481 

necessarily  loss  than  a,  it  follows  that  these  remainders  include  all  numbers 
from  1  to  a  —  1. 

417.  Hence  again,  it  appears  that  some  one  of  the  above  terms  will  leave 
a  remainder  1  ;  aniHhat,  therefore,  if  b  and  a  be  any  two  numbers  prime  to 
each  other,  a  number  .t<>  may  be  found  that  will  render  hx—1  divisible  by 
a,  or  the  equation  6a: — ay  =  ~L  is  always  possible  if  a  and  b  are  numbers  prime 
to  each  other. 

And  it  is  always  impossible  if  rf  and  b  have  any  common  measure,  as  is  evi- 
dent, because  one  side  of  the  equation  bx — ay  =  ^  would  bo  divisible  by  this 
common  measure,  but  the  other  side,  J,  would  not  be  so;  therefore,  in  this 
case  the  equation  is  impossible. 

418.  If  a  be  any  prime  number,  then  will  the  formula 

1.2.3.4.5,  &c.,  (a  — 1)  +  1 
be  divisible  by  a ;  for  it  is  demonstrated  in  our  preceding  second  deduction, 
that  if  a  and  b  be  any  two  numbers  prime  to  each  other,  another  number  x 
may  be  found  <[«,  that  renders  the  product  bx — l-^-*a,  or,  which  is  the 
samo  thing,  &ar=ya+l  ;  and  that  there  is  only  one  such  value  of  x<C.a,  may 
be  shown  as  follows  : 

The  foregoing  equation  gives,  by  transposition, 

bx — ay  =  l ; 
and,  if  it  be  possible,  let  also 

bx' — ay'=l ; 
tnd  make  x'=x±m  and  y'=.y^zn,  where  m  is  necessarily  less  than  a,  be- 
cause bother  and  x'  are  so  by  the  supposition. 
Now,  by  this  substitution,  we  have 

(bx^zbm)  —  (ay^zan)  =  l ; 
DUt  /;./•  — «]/  =  l  ; 

therefore  ^:bm  =  :^:an,  or  bm-^-a ;  but  this  is  impossible,  since  b  is  prime  to 
«  and  m<a,  as  in  Art.  415.     There  can  not,  therefore,  be  two  values  of  x  less 
\han  a,  that  render  the  equation  6.r — oy  =  l  possible. 
But,  in  the  series  of  integers 

1  .2.3.4.5 a— 1, 

eveiy  term  is  prime  to  a  except  the  first,  a  being  itself  a  prime  ;  if,  therefore, 
we  write  successively  6=2,  b'  =  ?>,  b"=4,  &c,  a  corresponding  term  x,  in 
the  same  series,  may  be  found  for  each  distinct  value  of  6,  that  renders  the 
product  xbzcay-\-l,  x'b'3zay'-\-l,  x"b"^zay"-\-l,  &c. ;  and  it  is  evident  that 
no  one  of  these  values  of  x  can  be  equal  either  to  1  or  a  —  1  ;  for,  in  the  first 
case,  we  should  have  1  x6=#2/+l>  which  is  impossible,  because  b<^a  ;  and 
the  second  would  give  (a  —  l)b=ay-\-l,  or  a(b — y)  =  b-\-l;  that  is.  b-\-l-±±a, 
which  can  only  be  when  b=a  —  1,  or  when  b=.c,  which  case  is  excepted,  be- 
cause we  suppose  two  different  terms  of  the  series.     In  fact,  since  (a — l)2 

ac  «?/-{- 1,  there  can  be  no  other  term  in  the  same  series  that  is  of  this  form ; 
for  if  x~32(iy'+l,  then  (a  —  1)"  — x"  would  be  divisible  by  a,  or  (a  —  l-\-x) 

X  (rt  —  1 — ,r)-H-a,  which  is  impossible,  since  each  of  these  factors  is  prime  to 

*  To  save  the  repetition  of  the  words  "  divisible  by,-'  which  frequently  occur,  the  sign 
^-  is  used  to  express  them ;  and,  for  the  same  reason,  the  symbol  n:  is  introduced,  to  ex- 
press the  words  "of  the  form  of,"  which  are  also  of  frequent  occurrence. 

Ha 


482  ALGEBRA. 

a,  as  is  evident,  because  .r<a,  and  a  is  a  prime  number.     Hence  cur  product 

1.2. 3. 4. 5. ...(a— 1) 

becomes  1  .bx.b'x'  .b"x" a  —  1; 

but  each  of  these  products,  bx,  b'x',  b"z",  &c.,  is,  as  wo  have  seen,  of  the 
form  ay-\-l ;  therefore  their  continued  product  will  have  the  same  form,  and 
the  whole  product,  including  1  and  a  —  1,  will  be 

a=  (ay+1)  X  {a-\)^a-y+ay+a-l, 
to  which  if  unity  be  added,  the  result  will  bo  evidently  divisible  by  a  ;  that 
is,  the  formula 

1.2.3.4.5 (a  — 1)  + 

is  always  divisiblo  by  a  when  a  is  a  prime  number. 

DEDUCTIONS. 

(1)  The  product 

1.2.3.4.5 («  — 1) 

is  the  same  as 

l(a-l)2(«-2)3(a-3),  &c.,  (^^f  5 

and  this  product,  as  regards  remainder,  when  divided  by  a,  is  the  same  aa 

±i-.22.3*.4* (^)  ; 

the  ambiguous  sign  being  -4-  when  a  —  1  is  even,  and  —  when  a  — 1  is  odd  ;  i.  e., 

-f-  when  a  is  a  prime  of  the  form  4n-{-l,  and  —  when  a  is  a  prime  of  the  form 

in — 1 ;  also,  this  last  product  is  the  same  as 

,   /  a— 1\2 

±(1-2.3.4 —)   ■ 

therefore,  from  what  is  said  abovo  relating  to  the  ambiguous  sign  we  shall  have 

l(l-2-3.4 ^-Zl)-+1^a 


when  azc4n-f-l  ;  and 


{('■•■■■« *-?)•-}* 


when  azcin  —  1. 

Hence  eveiy  prime  of  the  form  4/i-f- 1  is  a  divisor  of  the  sum  of  two  square! 
Again,  the  latter  form  may  be  resolved  into  the  two  factors 


1.2.3.4... 


¥)-?• 


which  product  being  divisible  by  a,  it  follows  thai  a  is  a  divisor  of  one  or  otho 

of  these  factors  when  it  is  a  prime  number  of  the  form  4fl  —  1. 

(2)  From  the  first  product,  which  we  have  shown  to  be  divisiblo  by  6,  v« 

1.2.3.4,  &c,  («_l)  +  l 
=c,  an  integer, 

we  may  derive  a  great  many  others,  as 

1«.2».3.4,  &c,  (a— 3)(a  — 1)+1 


=c,  an  intejror, 
a  ° 

P. 2*. 3'. 4. 5,  <kc,  (a—4)(a  — 1)+1 
=r,  an  integer, 

and  so  on  till  wo  arrive  at  the  same  form  as  that  in  the  tir>t  deduction 


PRIMITIVE  ROOTS.  483 

PRIMITIVE  ROOTS. 

419.  Theorem. — If  p,  a  number  prime  to  a,  divide  the  successive  powers  1, 
b,  a2,  a3  .  .  .  there  will  be  one  at  least,  before  arriving  at  aP,  which  ivill  leave 
the  remainder  1. 

The  remainders  being  each  less  than  p,  there  can  be  but  p  —  1  different 
ones,  and,  therefore,  in  the  p  first  terms  of  the  series  1,  a,  a~,  a3  .  .  .  ap_1, 
there  are  at  least  two  which  will  give  the  same  remainder.  Representing 
them  by  am,  am',  and  their  common  remainder  by  r,  suppose 

am  =  Ep-f  r,  a'n'=E'p-\-r (1) 

...  am'— am  =  (E'— E)^,  or  am(am'-m  —  1)  =  (E'  —  E)p; 
and,  as  p  is  prime  to  a,  it  must  divide  am'-m  —  1.     Therefore  we  have  unity 
for  remainder  in  dividing  by  p  the  power  am'~m,  which  is  <a".     Q.  E.  D. 

420.  Let  an  designate  the  lowest  power  other  than  a0,  which  gives  tho  re- 
mainder 1.  All  the  preceding  remainders  are  unequal.  For,  if  for  two 
powers,  am,  amr  less  than  an,  we  could  have  the  equalities  (1),  we  might  con- 
clude, as  just  now,  that  a""~m  would  give  the  remainder  1.  Consequently, 
an  would  not  be  the  lowest  power  to  which  this  property  belonged. 

THEOREM    OF    FERMAT. 

421.  If  p  be  a  prime  number  which  will  not  divide  a,  the  division  of&V'1  by 
p  will  give  1  for  a  remainder.  In  other  words,  ap_1  —  1  is  exactly  divisible 
hyp. 

It  must  be  carefully  observed  that  p  is  an  absolute  prime  number,  and  not 
simply  prime  to  a. 

Call  q,  q',  q",  .  .  .,  and  r,  r',  r",  .  .  .  the  quotients  and  remainders  of  the 
p — 1  quantities  a,  2a,  3a  .  .  .  (p  —  l)a,  divided  by  p.  If  we  multiply  these 
quantities,  and  suppose  E  to  be  an  entire  number,  we  have 

a.  2a.  3a {p  —  l)a=(qp  +  r){q'p  +  r'){q"p+r")  .  .  . 

=E+rr'r"  .  .  . 

The  first  member  is  equal  to 

1.2.3....  (p— l)aP-' 
and,  as  the  remainders  r,  r',  r"  .  .  .  are  all  different  (Ait.  415),  the  product 
rr'r"  .  .  .  must  evidently  be  that  of  the  whole  series  of  natural  numbers,  1,  2, 
3  ,  .  .  (*j  —  i),  from  1  to  (p  —  1).     Hence  the  above  equality  becomes  • 

1.2.3 {p—l)xaP-1=Kp  +  l  .2.3...  (p— 1) 

.-.  1.2.3...  {p  —  l)(aP-'  —  l)=Ep. 

Tho  1°  member  of  this  equality  is,  therefore,  divisible  by  p  ;  but  since  p  is 
a  prime  number,  it  can  not  divide  any  of  the  factors  1  .  2  .  3  .  .  .  (p— -1)  ;  it 
must,  therefore,  divide  a?-1  — 1.  Q.  E.  P. 

Suppose  that  we  take  for  p  only  prime  numbers  ;  if  we  wish  that  the  pow- 
ers an,  a1  .  .  .  ap-1  should  give  for  remainders  all  the  numbers  inferior  to  p,  it 
is  necessary  to  choose  a,  such  that  aP_1  should  be  the  lowest  power  above  a°, 
which  gives  the  remainder  1  ;  and  if,  among  those  which  fulfill  this  condi- 
tion, we  take  for  a  only  numbers  below  p,  Ave  have -those  which  Euler  calls 
primitive  roots. 

For  the  best  method  of  calculating  them,  the  student  is  referred  to  the 
article  by  Mr.  Ivory,  in  the  fourth  volume  of  Supplement  to  Encyclopedia 
Britaunica.  We  shall  limit  ourselves  to  setting  down  hero  the  primitive  root* 
of  numbers  as  far  as  37. 


484 


ALGEBRA. 


Numbers  p. 


Primitive  roots  of  p. 


3 

o 

5 

o 

3 

7 

3  . 

5 

11 

2 

6 

.  7 

.  8 

13 

2 

6 

.  7 

.  11 

17 

3  . 

5 

.  6 

.  7 

.  10 

.  11 

.  12  .  14 

19 

2 

3 

.  10 

.  13 

.  14 

.  15 

23 

5  . 

7 

.  10 

11  . 

13 

14 

15  .  17 

29 

2  . 

3 

.  8 

10  . 

11  . 

14  . 

15  .  18 

31 

3  . 

11 

12 

13 

17. 

21  . 

22  .  24 

37 

2 

5 

.  13 

15  . 

17. 

18  . 

19  .  20 

20  .  21 
19  .  21 


26.  27 


22  .  24  .  32  .  35 

THE  FORMS  OP  SQ.UARE  NUMBERS. 

422.  Every  square  number  is  of  one  of  the  forms  An  or  4»-{-l. 

Every  number  is  either  even  or  odd ;  that  is,  every  number  is  of  one  of  the 
forms  2n  or  2n-\-l;  and,  consequently,  every  square  is  of  one  of  the  forms 

4»s:n4n, 
4n2+4n+l3z4?i+l. 

DEDUCTIONS. 

(1)  Every  even  square  number  is  divisible  by  4. 

(2)  Since  every  odd  squaro  by  the  above  is  of  the  form  4(n--{-n)-{-l,  and 
since  n2-4-tt  is  necessarily  oven,  it  follows  that  every  odd  square  is  of  the  form 
8n-f-l;  and,  consequently,  no  number  of  the  forms  8n-{-3,  Sn-\-5,  8n-\-7 
can  be  a  square  number. 

(3)  The  sum  of  two  odd  squares  can  not  be  a  squaro  ;  for 

(8w+l)  +  (8/i+l)^:4n  +  2) 
which  is  an  impossible  form. 

423.  Every  square  number  is  of  one  of  the  forms  5re  or  5nil.  For  all 
numbers,  compared  by  the  modulus  5,  are  of  one  of  the  forms 

5ra,  5«il?  5ttrb2 : 
and  all  squares,  therefore,  are  of  one  of  the  forms 

25n2  =c5«, 

25ft'±10w4-l3=5«4-l, 

25n2+20rt+4  3:5n+4  or  on  —  1. 
Therefore  all  squares  are  of  one  of  tin;  forms  5«  or  5?j±1. 

DEDUCTIONS. 

(1)  If  a  square  number  be  divisible  by  5,  it  is  also  divisible  by  25 ;  and  if  a 
number  be  divisible  by  5  and  not  by  25,  it  is  not  a  square. 

(2)  No  number  of  the  form  5n-\-2  or  5h-|-3  is  a  square  Dumber. 

(3)  If  the  sum  of  two  squares  be  a  squaro,  one  of  the  three  is  divisible  by 
5,  and,  consequently,  also  by  25  ;  for  all  the  possible  combinations  of  the  three 
forms  57?,  5n-\-\,  and  on  —  1  are  as  follows  : 

(5„  4-1)4.  (5n' + 1)  a:  on  4-  2, 

(5n  — 1)  +  (5»'— l)=c5n— 2nr5tt4_:;. 

5n        4"  ,r,rt'         =c5n, 

5„         4_(-,„'4_i)-i--)„_4_i, 

gfl  +(on'  —  l);n5/i  —  1, 

(r„,-|_i)4-(.-,«'_i)^:5«. 


FORMS  OF  SQ.UA11E  NUMBERS.  485 

Now,  of  these  six  forms,  the  latter  four  have  one  of  the  squares  divisible  by 
5,  and,  therefore,  also  by  25.  And  the  first  two  are  eacli  impossible  forms 
for  square  numbers  ;  that  is,  neither  of  thcso  two  combinations  can  produce 
squares ;  thorofore,  if  the  sum  of  two  squares  be  a  square,  one  of  the  three 
squares  is  divisible  by  25. 

(4)  In  a  similar  way,  it  may  be  shown  that  all  square  numbers  compared  by 
modulus  10  are  of  one  of  the  forms 

lOn,  10ra  +  5,  10/1+1,  10rc+6,  10;i+4,  or  10n+9. 
Therefore,  all  square  numbers  terminate  with  one  of  the  digits  0,  1,  4,  5,  6, 
or  9 ;  and  hence,  agaiu,  no  number  terminating  with  2,  3,  7,  or  8  can  be  a 
square  number. 

(5)  By  examining,  in  like  manner,  the  forms  of  squares  to  modulus  100,  we 
may  deduce  the  following  properties  : 

(6)  A  square  number  cau  not  terminate  with  an  odd  number  of  ciphers. 

(7)  If  a  square  number  terminate  with  a  4,  the  last  figure  but  one  must  be 
even. 

(8)  If  a  square  number  terminate  with  a  5,  it  must  terminate  with  25. 

(9)  If  the  last  digit  of  a  square  be  odd,  the  last  digit  but  one  must  bo  even  : 
and  if  it  terminate  with  any  even  digit  except  4,  the  last  but  one  must  be  odd. 

(10)  A  square  number  can  not  terminate  with  inoro  than  three  equal  digits, 
unless  they  are  0's ;  nor  can  it  terminate  with  three,  unless  they  are  4's. 

424.  All  square  numbers  are  of  the  same  form  with  regard  to  any  modulus, 
<z,  as  the  squares 

02,  l2,  22,  33,  &c.  (ia)3,  a  being  even ; 
and  as 

0s,  l2,  22,  32,  &c.  y-^-)  ,  a  being  odd. 

For  eveiy  number  may  be  represented  by  the  formula  an^r,  in  which  / 
shall  never  exceed  \a. 

Now  (cm±r)"=a2n'iA:2arn-^i-2, 

where  it  is  obvious  that  r"  and  (an^r)2  will  leave  the  same  remainder  when 
divided  by  a;  therefore,  (an^r)"2  and  r2  will  be  of  the  same  form  compared 
by  modulus  a  ;  but  r  never  exceeds  \a,  therefore  all  numbers  compared  bv 
modulus  a  are  of  the  same  forms  as 

0s,  l2,  22,  32,  &c,  r2, 
or,  as  the  squares, 

02,  l2,  22,  3s,  &c,  {\af,  when  a  is  even, 
and  as 

03,  l3,  22,  32,  &c,  \— — J  ,  when  a  is  odd. 

DEDUCTIONS. 

(1)  When  a  is  even,  the  general  formula 

(/:/r:rt:2anr-j-r2 
becomes  4a'2n3i4a'nr-f-r2 

3r4a'(a'ra3±«r)-f-r2. 
Therefore,  all  square  numbers  are  of  the  same  form  to  modulus  4a  as  the  squares 

02,  l2,  23,  32,  &c.,  a" ; 
and  hence  we  see  immediately  that  all  square  numbers  to  modulus  8  must  bf> 
of  the  same  fonns  as  the  squares 


486  ALGEBltA. 

0:,  1«,  2s 

that  is,  they  are  all  of  the  form 

8ra,  8ra-fl,  8n+4, 
as  we  have  already  demonstrated. 

(2)  The  following  tables  exhibit  the  possible  and  impossible  forms  r*  square 
numbers  for  all  moduli  from  2  to  10. 

Possible  Formula. 
2n,     2ra+l, 
3n,     3n  +  l, 
An,     4  7i  + 1 , 
5ra,     hn^-l, 

6n,     6w+l,     6?i  +  3,     6?i+4, 
7n,     7n+l,     7rc  +  2,     7n-j-4, 
8w,     8«-fl,     8re+4, 
9n,     9ra+l,     9n  +  4,     9/i  +  7, 

10«,  10n±l,  10n±4,  lOwio 

Impossible  Formulce. 
3«, 

4«,  4«  +  3, 

5n,  5n-j-3, 

6n,  6n -f-^i 

7ra,  7n-f5,  7«  +  G, 

8rc,  8n±3,  Sra-f.7, 

9n,  9?i±3,  9«-f  5,  9«+8, 

lOrc,  lO/iztS. 


CONTINUED  FRACTIONS. 
425.  Tue  name  continued  fraction  is  given  to  an  expression  of  the  form 
1    i  1 

+4+i    i  U+i     i 

64--  c4-- 

^8  «'  +  ,  &c-i 

l.  c,  a  fraction  whose  denominator  is  a  whole  number  and  a  fraction,  and 
which  latter  fraction  has  also  for  its  denominator  a  whole  number  plus  a  frac- 
tion, and  so  on. 

An  expression  whoso  numerators  and  denominators  are  any  quantities  what- 
ever, may  have  the  form  of  a  continued  fraction  ;  but  continued  fractions,  of 
which  the  numerators  are  1  and  the  denominators  whole  positive  numbers,  are 
the  kind  which  most  usually  occur. 

These  expressions  arise  in  various  ways,  and  are  of  great  use  in  finding  (lit; 
approximate  values  of  fractions  and  ratios  that  are  expressed  in  large  numbers, 
as  well  as  in  the  resolution  of  certain  unlimited  problems  of  the  first  and  second 
degrees:  in  the  latter  of  which  the  answer  can  not  be  easily  obtained  in  whole 
numbers  by  any  other  method. 

Thus,  in  order  to  represent  tile  irreducible  fraction  or  ratio     by  a  continued 

I) 


CONTINUED  FRACTIONS.  487 

fraction,  let  b  bo  contained  in  a,  p  times  with  a  remainder  c  ;  also,  let  c  be  con- 
tained in  b,  q  times  with  a  remainder  d,  and  so  on,  according  to  the  following 
scheme  : 

b)  a  (p 

c)_Hq 
d)  c   - 
e)  d  (s 
f,  &c, 
and  we  shall  have,  by  the  principles  of  division, 

a  c   b  d    c  e 

c  d 

p,  q,  r,  &c,  are  called  partial  quotients,  and  p-\-r,  q-\-~ »  &c,  complete 

quotients. 

By  taking  the  reciprocals  of  the  second,  third,  &c,  of  the  above  equations, 
we  have 

£_1  d_l 

.:%=P  +  l=p  +  -+d=p+l-      1 

Whence,  by  extending  the  number  of  terms  and  generalizing  the  formula,  wo 
shall  have 

a  1  a      1 

7=P+~  .  1  or  t=—  ,  1  « 

b  q-\ —      1  b     p4--      l 

According  as  the  numerator  is  greater  or  less  than  the  denominator ;  for  in  the 
latter  case  we  should  invert  the  first  as  well  as  the  second,  third,  &c,  equations. 
To  convert  a  given  irreducible  fraction  into  a  continued  one,  we  have  the 
following 

RULE. 

Divide  the  greater  of  the  two  terms  of  the  fraction  by  the  less,  and  the  last 
divisor  continually  by  the  last  remainder,  till  nothing  remains,  as  in  finding  their 
greatest  common  measure  ;  then  the  successive  quotients  thus  found  will  be 
,.ho  denominators  of  the  several  terms  of  the  continued  fraction,  the  numera» 
tors  of  which  are  always  1. 

EXAMPLES. 

2431 
(1)  Reduce  .  _-,  to  a  continued  fraction. 
*  '  1051 

1051)  2431  (2 
2102 

329)  1051  (3 

987 


64)  329  (5 
320 


9)  64  (7 
63 


1)  9  (9. 


488  A!- 

2431  1 

Hence  1051  —  +  3  +  1-      1 

5+-      1 
7+-. 

(2)  1096_1 

9119  =  8+I     1 

(3)  421_1 

<I72— -2+-      1 

As  the  fracticn  -,  in  every  case  of  this  kind,  is  supposed  to  be  irreducible,  or  in  its  low- 
b 

eat  terms,  it  is  evident,  by  following  the  above  process  (which  is  similar  to  the  method 
used  for  finding  the  common  measure  of  true  numbers),  that  we  shall  necessarily  arrive  at 
a  remainder  equal  to  1 ;  or  otherwise  a  and  b  would  have  a  common  divisor,  which  is  con 
trary  to  the  hypothesis. 

Tho  continued  fraction  obtained  will  consist  of  a  greater  or  less  number  of  terms,  accord- 
ing as  the  fraction  -  is  more  or  less  complicated ;  but  they  will  always  terminate  when  - 

\s  rational. 

426.  A  continued  fraction  may  be  converted  into  a  series  of  vulgar  fractions 
by  finding  the  successive  sums  of  its  several  terms,  after  the  manner  of  redu- 
cing complex  fractions  to  simple  ones,  in  common  arithmetic ;  and  the  result 
will  be  more  or  less  accurate,  according  to  the  number  of  terms  of  the  con 
tinued  fraction  employed. 

Each  of  these  results  is  called  a  convergent,  and  they  are  numbered  in 
order. 

Thus,  if  it  were  required  to  reduce  the  following  continued  fraction, 

1 

H —     1 

C+-,  &c, 

to  a  series  of  common  vulgar  fractions,  the  operation  will  stand  thus : 

a  .  1     ab-\-l  ,  1  c  abc4-a-\-c 

«=I  (1),  a  +  l=-±-  (2),  fl+-+l=a+_=_J!lJ^ 


or 


(ab  +  l)c+a  1  1  cd+1 

— ^-p—  (3),a+j     1      !=*+£+     rf=«+£rf+&+2 

C-f--.  ('(/-)- 1 

abcd-\-  ab  +  ad+cd+l      [{ab  +  l)c+a]d-\-tib  +  l 

(4) 


—  bcd+b+d  ~  (oc-fl)</+6 

(1),  (2),  (3),  and  (4)  are  called  the  first,  second,  third,  and  fourth  convergent. 
427.  By  inspecting  tho  above  convergent^,  we  perceive  that  each  may  be 
formed  from  tho  preceding  by  the  following 

nui.r.  § 

Add  the  product  of  tho  numerator  of  the  convergent  already  found  by  tht> 
denominator  of  the  next  term  of  the  continued  fraction,  to  the  preceding 


CONTINUED  FRACTIONS.  489 

numerator,  for  the  next  numerator  and  follow  the  same  process  for  the  de 
nominator.* 

EXAMPLE  I. 

1 

3+5+i     1 

denominators  or  quotients  3,    5,    2,      7,  arranged  in  horizontal  line , 

3   16   35  261 
convergents  ^  "F>  JT'  "go"' 

3         16 
After  having  formed  the  convergents  -  and  — ,  the  rule  applies.     Then  mul- 

tiply  16,  the  second  numerator,  by  2,  the  third  quotient,  and  add  3,  the  pre- 
ceding numerator,  it  gives  35 ;  and  multiplying  5,  the  second  denominator,  by 
the  same  quotient  2,  and  adding  1,  the  preceding  denominator,  it  gives  11  ; 
and  so  on.     This  method  may  proceed  from  the  commencement,  if  we  write 

-  before  the  first  convergence. 
Thus, 


3, 

5, 

7, 

1 

3 

16 

35 

261 

0 

1 

5~ 

11 

~82~ 

"When  the  continued  fraction  is  not  terminated,  the  numerators  and  denom- 
inators form  two  series  increasing  to  infinity. 

428.  The  convergents  are  alternately  less  and  greater  than  the  value  of  the 
continued  fraction ;  for  the  first  in  the  general  form  is  equal  to  a,  and  as  the 
fractional  part  which  is  added  is  neglected,  this  is  too  small.     The  second 

convergent  is  a-{-7'  *md,  sinco  b  is  too  small  by  -,  the  fraction  7  is  too  great, 
and,  consequently,  the  whole  convergent ;  and  so  on. 

EXAMPLE  11. 

It  is  shown  in  geometry  that  the  ratio  of  the  circumference  of  a  circle  to 

31415926535         . 
its  diameter  is  j „»..».  „..»..»,  which,  by  being  converted  into  a  continued  frac 

tion,  and  the  successive  convergents  found,  will  be  as  follows  : 

3  22  333   355   103993 
1'  7"'  106'  113'   33102'  ;t 

*  The  generality  of  tbis  rule  may  be  proved  as  follows : 

N  N'  N" 
Let  — ,  — ,  — ■  be  three  consecutive  convergents,  m  the  quotient,  of  the  same  rank  as 

N"  1  1 

the  convergent  — -,  and  -  the  partial  fraction  which  follows  — ;  and  let  N"=N'w-f-N  and 
D"  n  m  ' 

Nv 
D"=D'm-f-D,  according  to  the  rule.     The  convergent  which  follows  —  is  formed  by  sub- 

1  N"  N'm-f-N 

stituthig  m-\ —  for  ?i  in  -— .     Making  this  substitution  in  its  equivalent  — — — ,  we  have 
n  1)  D/n-j-D 


N'  (m+-)  +N     rtT/ 


(N'm+NJn+N'     N"»+N' 


22 
t  Tlie  second,  — ,  was  the  ratio  assigned  by  Archimedes ;  the  third,  which  is  mucu 

more  accurate,  that  by  Metius 


490  ALGEBRA. 

and  either  of  these  will  be  the  approximate  value  of  the  ratio,  more  and  more 
accurate  as  we  advance. 

429.  The  difference  between  two  convergents  is  equal  to  1  divided  by  the 
product  of  the  denominators  of  the  two  convergents.     Thus,  in  the  above  ex 

ample,  the  difference  between  the  first  and  second  convevgents  is  -,  hetwenn 

the  second  and  third  it  is  - — — -,  or  — — ,  between  the  third  and  fourth  ■  , 

and  as  the  true  value  of  the  continued  fraction  is  somewhere  between  any 
two  consecutive  convergents,  we  have  its  value  to  within  less  than  the  fraction 

?>  ^7o'  or  nn-rc.'  &c#'  according  to  the  convergent  which  we  take. 
/    74J         llU7o 

To  prove  this  in  a  general  way,  let 

N  N'  N" 
D'  IT"  D7' 

be  three  consecutive  convergents,  and  m  the  quotient,  of  the  same  rank  as  the 

N" 
convergent  t—  ;  then  N"  =  N'm-|-N ;  D"=D'»i-|-D. 

N'     N     DN'— DN 
But  D-D=— BW- W 

N"     N'     N'm+N     N'_N'D'm+D'N-N'D'm-DN' 
'*  W'~ D7==D'7ft+D"-D7=_  D'(D'm  +  D) 

D'N— DN'      D'N— DN7 
"~ D'(D'm  +  D)  —      D'D~  '  '  ^ 

The  numerators  of  (1)  and  (2)  are  the  same,  with  contrary  signs  ;  and,  to 

a 
find  its  value,  we  have  only  to  go  back  to  the  first  two  convergents  -  and 

rtfc  +  1  1 

— r — i  the  difference  of  which  is  7. 
0  b 

430.  Since  the  denominators  of  the  convergents  increase  to  infinity  if  tlie 
series  continue  sufficiently  far,  it  is  possible  to  take  two  consecutive  convergents 
whose  difference  shall  bo  less  than  any  assignable  number  ;  wherefore,  as  two 
consecutive  convergents  comprehend  between  them  the  value  of  the  continued 
fraction,  it  follows  that  a  convergent  can  be  found  whoso  value  shall  ciller  from 
that  of  the  fraction  by  less  than  any  assigned  number. 

For  example,  if  the  valuo  of  a  continued  fraction  be  required   to  within 

,  the  convergents  must  bo  continued  till  the  product  of  tho  denominators 

of  the  last  and  last  but  one  is  at  least  3  000.  The  last  convergent  will  then 
have  the  degree  of  approximation  required. 

N 
The  convergents  aro  fractions  in  the  lowed  terms  ;  for  if  a  converge 

admits  of  lower  terms,  some  quantity  </  must  be  a  cmiiimii.ii  measure  of  N  and 
D.  Whence  ( \rt.  29)  q  must  be  a  measure  of  the  multiples  N'D  and  N  I  >  . 
and  of  ( \rt.    •-".»)  DN'  —  ND    or  i  1 ,  which  is  impossible. 

431.  Each  convergent  is  n  nearer  approximation  to  the  tree  value  of  the 


CONTINUED  FRACTION'S.  491 

N"      N'?/j  -4-  N 
tmued  fraction  than  tbnt  which  precedes.     For,  let  7777=-^ — r~r:  he  a  cmver- 

D        Dm-\-D 

gent  in  which  m  is  the  last  quotient  employed  ;  then,  if  the  complete  quotient 

M-|---{-)  &c.    30  denoted  by  y,  and  y  bo  substituted  for  m  in  the  expression 

N" 
of  y—,  it  is  evident  (employing  x  to  denote  the  value  of  the  continued  fraction; 

that 

N'y+N 

N   N' 
Subtracting  each  of  the  convergents  =r,  r--  from  this  value  of  x, 

N'y+N     N     (DN'— ND')//  ±y 


D'y+D     D~~  D(D'y  +  D)  ~"D(D'y  +  D)' 


N'y+N     N'     ND'— DN'  qpl 

•nd 


D'y+D     D'— D'(D't/  +  D)  — D'(D't/+D)* 
But  2/>l  and  D'>D  .-.  D'(D'y+D)>D(D'7/+D) ; 


?/        ;>         ' 


••D(D'3/  +  D)^D'(D'7/+D)- 

N'  N 

vVhence  ■=?,  is  a  nearer  approximation  to  the  value  of  a;  than  =r. 

432.  Among  continued  fractions  those  have  been  particularly  distinguished  in 
which  the  denominators,  after  a  certain  number  of  changes,  are  continually 
•epeated  in  the  same  order,  in  which  the  continued  fraction  so  formed  is  said 
to  be  periodic,  and  may  then  always  be  considered  as  the  root  of  a  quadratic 
equation  or  a  surd. 

To  prove  this,  take  a  continued  fraction  entirely  periodic. 

1 
xs=-     1 
p+-     1 

PJrp+,  &c. 

Then,  since  the  number  of  these  fractions  is  unlimited,  it  follows  that  the 

sum  of  all  after  the  first  is  also  .r  ,-  whence 

1 
.r= — ; —  .■.  ar3+«.r=l 
p-\-x  '  * 

•••  x=— 5i»±-J  Vp*+4\ 
in  which  case  the  above  continued  fraction  serves  to  determine  tr  e  vahie  of 
v^p2_|_4)  since  we  have,  by  transposition, 

P     1 
-4—     l 

2^>+-      1 

P+p+,tcc; 
and  if  p  in  this  last  expression  be  put  equal  to  2,  we  shall  have 

1 


lVp*+*=lp+z=%+: 


V2  =  l+-      1 

v  ~2_l_-      l 

^2  +  -      l 

2-L- 

^2+,  &c 
A  continued  fraction  is  also  called  periodic  when  the  denominators  occur 
periodically  in  pairs,  threes,  fours,  &e. ;  thus, 


492  ALGEBRA. 

1  1 

-      1  or-     ] 

P+-     1  P+-     1 

9+-  1  ?+-      1 

P+q+,&C.  r+p+-1      I 

*+f.  ' 

Again,  a  fraction  may  be  irregular  in  some  of  its  first  terms,  and  only  become 
periodic  at  a  certain  distance  from  its  commencement. 

In  either  of  these  cases,  as  above,  the  value  of  x,  the  sum  of  all  the  terms, 
may  be  obtained  by  the  resolution  of  an  equation  of  the  second  degree.  To 
prove  this  in  a  general  manner,  let 

a,  o, . . .  .  &c.,  be  the  quotients  which  form  the  non-periodic  part, 
p,  q, . .  . .  &c,  be  the  quotients  which  form  the  periodic  part ; 

then  x=a-\—r 

":      1 
+-     1 

P+q+,  &c.J 

and,  representing  by  y  the  value  of  the  periodic  part, 

1 
P~*"q+,  &c, 

we  have  xz=a-\-r  and  y=p-\-- 

':       1  ':       1 

+  "  4— 

y  y 

Consider  these  continued  fractions  as  terminating  with  the  partial  fraction 
-,  and  deduce  the  convergents ;  we  have  (Art.  426)  two  equations  of  the  fol- 

k/ 

lowing  form  : 

_Py+P         R'y+R 

The  value  of  y,  given  by  the  first  of  these  equations,  is 

P-Q-r 
y  —  Q'.r— P" 
which  substituted  in  the  second,  gives,  after  reduction, 
P— Qx      R'(P  —  Q,r)-r.R(Q',r— P') 

Q'x_P'-S'(P_Q.r)  +  S(Q'.x-P')' 
which  is  an  equation  of  the  second  degree  in  x. 
By  way  of  illustration,  take  the  following  fraction  : 

V  V 

x=a+-  ,  p  (1)      or  .r— a=-  ,  v  (2) 

+«+f+£,to.  «+*+.  &«. 

p  2a — (7+  ■v/</"-M/> 

.-.  x — a= — ; ;  or,  resolving  the  equation,  x= — 

q  -\-x — a  '  2 

2a 
If  we  transpose  —  or  a,  and  substitute  for  x — a  its  value  (2),  we  have 


vV-+4p— q_  P 

■     2  ?+?.   /'    , 

q  +    .  &c.; 

or,  making  q  =  2a, 


CONTINUED  FRACTIONS.  4<J3 

P 


2a,  &c 
A  similar  mode  of  solution  may  be  applied  to  continued  surds  or  expressions 
of  the  form 


V«+V< 


J* 


a+  Vai  ^Cc*' 

the  value  of  which,  though  apparently  infinite,  is  always  determinable  by  a  cer- 
tain equation,  and  in  some  cases  in  a  real  integral  or  fractional  quantity  ;  for, 
putting 

x=\] a-\-~\Ja-\-  -/a,  &c-» 
we  shall  have,  by  squaring  both  numbers, 

u*=a-\-yl a+  Va+,  &c, 
the  latter  part  of  which  is  evidently  equal  to  the  original  surd ;  whence 

x2=a-\-x,  or  x- — x=a  .-.  x=\^z  -/{  +  «, 
where,  if  a=2,  the  expression  becomes 

r2+V2+  V2+,  &c.,  =2  or  —1. 
433.  The  process  for  developing  any  quantity,  x,  in  a  continued  fraction, 

1,1  1 

consists  in  making  successively  x=a-{-—,  x'  =  b-\-—,  x"=zc-\-—,  &c,  a  be- 
ing the  greatest  whole  number  contained  in  .r,  b  the  greatest  whole  numher 
contained  in  x',  c  the  greatest  whole  number  contained  in  x",  &c. 

The  numbers  a,  b,  c,  &c.,  being  found,  it  is  evident  that  if  x',  x''    &c.,  are 
replaced  by  their  values,  the  required  development  is  obtained,  viz. 

1 
x=a4-T  ,  1 

d-\-,  &c 

EXAMPLE. 

Let  it  be  required  to  convert  -\/19  into  a  continued  fraction. 

V~19=4+i  ...  ,-'=  -JL—  _  V^+4  , 
x  V19-4-         3     J_ 

'^£±1-^1-0 + i.  .  3       .     V19+2 

rty  proceeding  in  this  way  we  shall  obtain  the  following  : 

0.1 —  I L o_i_ —  . 

x—         3 -+-2.n. 

a/19 +  3  1 

rIII 1..        ' Q_L 

—        2        —     '  xlv ' 
*  Multiplying  both  numerator  and  denominator  by  i/19-M- 


494  ALGEBRA. 


V19  +  3  1 

riv -L 1_| 

x     —       c       —  i_r,.v 


V19  +  2  1 

— —  =  2-| 

3 ^V 

1 


,V 1 _L 0_L. 


xVI  = -^19  +  4=8+ 


XV1I1 


VH3  +  4  1 


Hence  V~19  =  4+-     1 

•   1+-     1 

3  +  1      l 

1+-      1 

•  VJI  being  tlio  same  as  r1,  it  is  evident  that,  omitting  the  4,  the  greatest  in- 
tegral part  of  -y/iy.  the  quotients  2,  1,  3,  1,  2,  8,  already  found,  wil  always 
return  again  in  the  same  order  to  infinity. 

Should  it  be  required  to  convert  the  square  root  of  19  into  a  series  of  con- 
verging fractions  without  first  reducing  it  to  the  continued  form,  they  may  be 
obtained,  after  the  method  before  employed,  from  the  integral  parts  of  the 
above  results  only. 

Quotients,  4,  2,  1,   3,    1,    2,     8,       2. 
1    4    9    13   48   61    170    1421 
6'  1'  2'   3~'  IT'  14'  ~39"'   326  ' 

EXAMPLES. 

Ans.  Quotients,  -,  22,   1,     4,      2. 
o 

1    22   23   114   251 
Convergents,  -,  -,  -,  — ,  — . 

Ans.  Quotients,  -,  7,    1,    2,     4,       5,        1,       2. 

1    7      8    23   100    iZZ     623    1769 
Convergents,  -,  -,  -,  -,  — ,  — ,  — ,  — . 

Ans.  The  quotients  are  5,  1,    1,    3,      5,       3,  tVc. 

5    6    11    39    206    657    . 
And  the  convergents,  -,  -,  j,  y,  — ,  —,  &C 

Ans.  The  quotients  are  5,    3,    2,      3,       10,  &c. 

5    16    37    127    1307 
And  the  convergents,  y,  — ,  —,  — ,  -gr— ,  Arc 

Ans.  Quotients,  6,  1,    2,     2,     2,        1,       12,  iVc. 

6    7    20    47    114    161    2046 
Convergents,  j,  j,  -,  y,  — ,  — ,  — . 

434.  The  converse  of  the  proposition  stated  in  Art.  432  is  true,  viz.,  that  tho 
root  of  an  equation  of  the  second  degree  may  be  expressed  in  functions  of  the 
coefficients  of  tho  equation  by  continued  fractions. 

The  general  form  of  tho  equation  of  tho  second  decree  may  be  wi  Ken 

ax-  —  bx— c=0 (1) 

in  which  b  and  c  may  be  essentially  negative.  This  may  be  put  under  the 
form 


(H 

251 
764* 

(2) 

1769 
5537' 

(3) 

t 

•/31 

(*) 

V28 

(5) 

V45 

CONTINUED  FRACTIONS.  4fl5 

,      c 
ax=b-\--. 

c 
Multiplying  the  fraction  -  above  and  below  by  a,  it  becomes 

ac  ac 

'ax  b  4-_     ac 

^  6+,&c. 


1/       ac 


"*"  6  +,  &c.J  Q.  E.  D 


If  a=l,  tliis  becomes 


If  6  =  0, 


0+6+,&c. 


x2=c, 


*=°+lLc- 

U  +  0  +  ,&c, 

which  has  no  signification.     But  if  we  make 

ar2=(z  — a)2, 

a3  being  the  greatest  square  contained  in  c,  we  have 

x2=z2 — 2az-}-a2=c,* 

••.  z2 —  2az  =  c — a2  ; 

or,  putting  c — a2=7, 

z2— 2az  =  y, 

z  -2a  =21, 

and  -  Z  =2°  +  L+,  &c. 

7 
But  since  x=z — a,  x=a-\-—       y 

~a-\- — ■ 

'  2a+,  &c. 

To  apply  this,  let  the  equation  be 

a:2=8  .-•  a=2,  7=4, 

•-=2+1     4 

^4  +  ,  &c, 
or 

+  1  +  ,  &c. 
The  above  result  may  be  obtained  in  a  more  simple  manner ;  tlnis,  put 
x2=c=o»4-/3  .-.  re2  — a2=/3  .-.  (x—a)(x-\-a)—3 

•'•  X=a+^+x=a+2-a+l 

T  T2o+,  &c, 

which  shows  that  the  square  root  of  any  number  which  is  the  sum  of  a 

square,  and  of  another  number,  is  a  continued  fraction. 


496  ALGEBRA. 

Thus,  if  we  have  .r-=7  .-.  o=2,  ,?=3, 

r-  3 

a/7=2+-     3 

M-f,  &c. 

435.  Continued  fractions  furnish  ;i  method  of  resolving  in  whole  numbeis 

the  indeterminate  equation 

ax-\-by=c (1) 

In  this  equation  a,  b,  c  are  whole  numbers,  and  the  first  two  are  supposed 

to  have  no  common  factor.     Let  us  conceive  that  we  have  developed  the 

a 
tion  r  into  a  continued  fraction,  and  that  we  have  calculated  all  the  con- 
o 

vergents ;  the  last  will  be  equal  to  this  relation  itself.     Let  us  subtract  from 

a' 
it  the  next  to  the  last,  which  I  represent  by  yr    The  numerator  of  the  differ- 
ence will  bo  ah'  —  ha',  and  by  the  property  of  Art.  430  we  have 

ab'  —  ba'=±l (2) 

Multiplied  by  +  <-',  this  equality  becomes 

aX±&'c+6x  faV=c; 
then  equation  (1)  is  satisfied  by  taking  ./.  •=  +  &'<:,  y=^za'c. 

This  solution  being  known,  we  Know  (Art.  1G1)  that  all  the  others  are  given 
Dy  the  formulas 

x=^zb'c — bt,  y=^a'c-\-al, 

t  designating  any  whole  number  whatever.     We  take  the  upper  or  lower  sign 

according  as  we  have  -f-  °r  —  in  the  equality  (2),  or,  what  is  the  same  thing, 

a 
according  as  the  convergent  7  is  of  an  even  or  uneven  rank. 

EXAMPLE. 

Let  there  be  the  equation 

261a:— 82y  ==117. 

261 
Tf  we  reduce  -77/  to  a  continued  fraction,  we  find 

Quotients,       3,    5,    2,     7. 

3   16   35   261 
Convergents,  -,  -j,  — ,  — . 

If  we  take  the  numerator  of  tho  difference  -  — — — ,  and  observe  that  77-7 

~ .'       11  82 

is  a  convergent  of  an  even  rank,  we  have 

261x11—82x35=4-1. 

Then,  multiplying  by  117, 

261  X  11  X  117  —  82  x  .".5  X  117  =  117. 
The  equation,  then,  is  satisfied  by  making  r=ll  x  117  =  1287  and  y=35 
Xll7=4096;  then,  finally,  the  general  values  of  x  and  y  aro 

x=1287+82«,  y==4095+261fc 
tf  we  divide  1287  by  82,  and  4095  by  261,  we  find  1287  =  82x15+57  and 
4095  =  261  xl">-f-180.     Then,  observing  that  t  is  any  whole  number  what- 
ever, we  can  write  more  simply 

r=574-82f,  y  =  180  +  261/. 


RATIONAL  SOLUTIONS  OF  QUADRATIC  INDETERMINATES.       497 

430.  The  following  theorem  will  be  found  useful  in  the  resolution  of  inde- 
terminate equations  of  the  second  degree. 

Let  p': —  A(/=iD  be  an  indeterminate  equation,  in  which  D<C  \/A.     1 

assert,  that  if  this  equation  is  resolvable,  tho  fraction  -  will  be  found  among 

the  (ructions  which  converge  toward  -/A. 

From  the  above  equation  we  derive  p — q-\/A  = =,  and,  therefore, 

p+qy/A 

V          —                                              rb<*              rfcD  D« 

-—  yf  A,  which  I  represent  by  — rr= — ;  then  d=- 


q  T       7(i;  +  (?VA)  p  +  qjA 

Let  —  be  tho  converging  fraction  which  precedes  -,  and  which  is  of  such  a 

nature  that  the  sign  of  6  will  bo  tho  same  with  that  of  D  ;  it  will  remain  to 

Dq  q  — 

be  proved  that  we  have —  <  — — , ovD(q-{-q0)<^p-i-q  ^  A. 

.p-\-qy/A      (l+% 

•      _      (5 
In  the  second  member,  instead  oi  p,  I  put  its  value,  qy/A^-;  tho   in- 
equality to  bo  proved  can  then  be  written  thus  : 

(7+7o)(  VA-D)  +  (<7-<7o)  VA±^>0. 

But  this  inequality  is  manifest,  sinco  we  have  VA>D,  q^>q0,  and  since 

the  part  (q—q0)  VA,  which  is  at  least  equal  to  VA,  by  itself  surpasses  -, 

•7 

which  is  less  than  unity.     — ,  then,  will  always  be  found  in  the  fractions  v, 

converge  toward  VA,  so  that  it  will  only  bo  necessary  to- develop  -/A  in  a 
continued  fraction,  and  to  calculate  the  converging  fractions  which  result,  in 
order  to  have  all  the  solutions  in  entire  numbers  of  the  equation 

x2— A2/2=iD, 
D  being  <  -/A. 

METHOD    OP    RESOLVING    IN    RATIONAL    NUMBERS    INDETERMINATE 
EQUATIONS  OF  THE  SECOND  DEGREE. 

437.  Let  the  proposed  general  equation  be 

ar  +  b.vy+cf+d.v+ey+f—O, 
in  which  x  and  y  are  the  indeterminates,  and  a,  b,  c,  d,  e,f\he  given  entire 
numbers,  positive  or  negative.     We  first  derive  from  this  equation  the  fol- 
lowing : 

2ax+by+d=V[(1>y+d)*-4a(cy*+ey+f)]. 
If   we    make,   to   abridge,    the    radical    =t,    i2— 4ac=A,    bd—2ae=g, 
d* — 4af=h,  we  shall  have  the  two  equations 

2<z.r-f   by-\-d=t, 
Ay*+2gy+h=zt*. 
If  we  multiply  the  last  of  these  equations  by  A,  and  make,  again,  Ay-\-g 
=i\  g3 — Ah=T>,  we  *ha!l  have  the  transformed  equation 

v2— At?=&. 
Reciprocally,  if  wo  can  find  values  of  v  and  t  which  satisfy  the  equation 

Ii 


498  ALGEBRA. 

v3 — A^=B,  we  deduce  from  it  the  values  of  the  inde terminates  x  and  y  in 
the  proposed  equation,  viz., 

v — g  t — by — d 

in  which  we  should  observe  that  both  v  and  t  may  be  taken  with  either  sign, 
as  we  may  desire. 

If  we  find  the  solution  of  the  proposed  equation  in  rational  numbers,  it  will 
■uffice  to  resolve,  by  means  of  these  numbers,  the  transformed  vz  —  A<:=B  ; 
but  if  we  wish  to  resolve  the  proposed  in  entire  numbers,  it  will  not  only  be 
necessary  that  t  and  v  be  entire  numbers,  but  that  the  values  of  t  and  v,  sub- 
stituted in  those  of  x  and  y,  give  for  these  indeterminates  entire  numbers.  At 
present  wo  will  only  occupy  ourselves  with  the  resolution  in  rational  numbers. 

438.  Every  indeterminate  equation  of  the  second  degree  can  be  reduced, 

as  we  have  just  seen,  to  the  form  v-  —  Af:=B;  but,  whatever  may  be  the 

rational  numbers  t  and  v,  wo  can  suppose  that  they  are  reduced  to  a  common 

x  y 

denominator.     Hence,    making   »==-,    t='-,  we   shall   have   to  resolve   the 

°  z  z 

equation 

.r2— Ai/2=B25, 
in  which  now  .r,  y,  z  are  entire  numbers. 

We  can  suppose  that  theso  three  numbers  have  not  a  same  common  divisor, 
for  if  they  had  had  one.  we  could  have  made  it  disappear  by  division. 

In  the  same  manner,  we  can  suppose  that  the  numbers  A  and  B  have  no 
square  divisors  ;  for  if  wo  had  had,  for  example,  A=A'Z;2,  B=B'l\  we  might 
have  made  J:y=y',  lz=z',  and  the  equation  to  be  resolved  would  have  become 

xt—A'y'—B'z'2, 
in  which  A'  and  B'  have  no  longer  a  square  factor. 

The  equation  a;3 — Ay':=Bz-  being  thus  prepared,  we  shall  observe  that  any 
two  of  the  indeterminates  x,  y,  z  can  not  have  a  common  divisor ;  for  if  #-,  for 
example,  should  divide  a:2  and  y",  it  must  necessarily  divide  also  Bz2 ;  but  it 
can  not  divide  ;:,  since  the  three  numbers  x,  y,  z  have  no  common  divisors ; 
neither  can  02  divide  B,  sinco  B  has  no  squaro  factor,  x  and  y,  therefore,  are 
prime  with  respect  to  each  other ;  for  the  same  reason,  x  and  z  are  primes 
with  respect  to  each  other,  as  well  as  y  and 

I  assert,  moreover,  that  A  and  B  can  be  supposed  to  be  positive  ;  for  we 
can  only  have,  as  regards  the  signs  of  the  terms  of  one  equation,  tho  following 
three  suppositions  : 

r=_A7-=4-l!:  . 
x*—Ay-  =  -iV. 
x2+Ay2=  +  B:. 

(I  omit  the  supposition  x--\-Ay"  = — B:-\  sinco  it  is  evidently  impossible.) 
Of  these  three  combinations  the  second  coincides  with  tho  third  hyn  simple 
transposition;  but  if  wo  multiply  the  third  by  B,  and  make*  B:=;',  AB  =  A\ 
we  shall  have 

z'-  —  A'y-=Bx': 
The  equation  to  bo  resolved,  therefore,  can  always  be  reduced  to  the  form 

x*—  By  '=  \ 
in  which  A  and  B  aro  positivo  numbers,  and  do  not  contain  any  square  factor 


RATIONAL  SOLUTIONS  OF  QUADRATIC  INDETERMINATES.       499 

439.  The  method  which  we  shall  proceed  to  follow  for  the  resolution  ot 
this  equation  is  that  given  by  Lagrange,  in  the  Memoires  de  Berlin,  1767.  It 
consists  in  producing,  by  means  of  transformations,  the  successive  diminution 
of  the  coefficients  A  and  B  until  one  of  them  becomes  equal  to  zero,  in  which 
case  the  solution  can  be  immediately  deduced  from  known  formulas. 

The  equation  thus  reduced  is  of  the  form  x" — y-  =  Az",  or  x2 — By*=z2 , 

but  these  two  formulas  do  not  differ,  and  it  will  suffice  to  give  the  solution  of 

the  first,  x- — y2=Az".     To  do  this,  decompose  A  into  two  factors  a,  ,8  (which 

will  always  be  prime  with  regard  to  each  other,  since  A  has  no  square  factor), 

and  suppose  that  z  also  is  decomposed  into  two  factors  p,  q,  such  that  we 

have  A=a/3,  z=pq,  we  shall  have  the  equation  (x-\-y)(x — y)=a[ip-q-,  which 

we  can,  in  general,  satisfy  by  taking  x-{-y=ap'2,  x — y=Pqi ;  this  supposition 

gives 

au2-f-/3<72           c/^—dq2 
x= ,  y= g -,  z=pq; 

hence  the  three  indeterminates  x,  y,  z  will  be  expressed  by  means  of  two 

arbitrary  quantities  p  and  q ;  if  it  should  happen  that  the  values  of  x  and  y 

contain  tho  fraction  A,  x,  y,  z  must  each  be  multiplied  by  two. 

Such  is  the  general  solution  of  the  equation  x\ — ?/2=Az2,  a  solution  which 

v.  ill  comprise  as  many  particular  formulas  as  there  are  ways  of  decomposing 

A  into  two  factors. 

For  example,  if  A=30;  there  are  four  ways  of  decomposing  30  iuto  two 

factors,  viz.,  1.30,  2.15,  3.10,  5.6  ;  hence  will  result  these  four  solutions  of  the 

equation  x* — t/2=30z2, 

1°.  ,r=  p2-f  30<72,  y==  p2—30q'2,  z  =  2pq, 
2°.  x=2jf--\-15q~,  y  =  2p'i—\bq2,  zz=2pq, 
3°.  x=3p~+10q2,  y  =  3p°-  — 10</2,  z  =  2pq, 
4°.  x=bp1-\-   6q2,  y=5p'2—   6q\  z  =  2pq. 

440.  Let  us  proceed  to  the  general  equation  x- — By2=Az2 ;  observe  that 
this  equation,  being  the  same  with  x* — Az2=By",  we  can,  without  diminish- 
ing the  generality  o/  the  theorem,  suppose  that  the  coefficient  of  the  second 
member  is  the  greater  of  the  two.  In  case  of  equality,  the  reduction  that  we 
shall  indicate  would  always  be  employed. 

Let,  then,  the  proposed  equation  be  .r2 — Bt/2=Az2,  in  which  we  suppose, 
at  the  same  time,  A>B,  A  and  B  positive,  and  free  from  any  square  factor. 

We  have  already  proved  that  x  and  y  are  primes  as  regards  each  other ;  ?/ 
and  A,  therefore,  are  equally  prime  to  one  another  ;  for  if  t/2  and  A  had  a 
common  divisor  6,  x*  also  must,  necessarily,  be  divisible  by  6,  and  x  and  y 
would  not  then  be  primes  to  one  another. 

But  since  y  and  A  are  primes  to  one  another,  if  we  suppose  that  the 
proposed  equation  is  resolvable,  and  that  we  can,  therefore,  find  determinate 
values  of  x  and  y,  .t=M,  y=^N,  we  shall  also  be  able  to  satisfy  the  equation 
of  the  first  degree, 

M=nN—  y'A, 
in  which  M,  N,  A  will  be  given  numbers  prime  to  one  another,  and  n,  y'  two 
indeterminates. 

Hence,  in  general,  without  knowing  the  particular  solutions  x=M,  y=N, 
we  can  suppose  x=ny — Ay'  n  and  y'  being  two  indeterminates;  and,  sub- 
stituting this  value  of  x  in  th*  proposed  equation,  we  shall  have,  after  having 
divided  by  A. 


500  ALGEBRA. 


/«-  — B\ 

\—L—)tf-2™ni'+Wi 


But  since  y  and  A  are  prime  to  one  another,  this  equation  can  not  subsist 

n=— B  ,    ■ 

unless,  — - —  be  an  entire  number.     Let  this  entire  number    =  A'k2,  k*  being 

the  greatest  square  which  can  be  a  divisor  of  it,  we  shall  have 

\'k-, 
and  the  equation  to  be  resolved  will  become 

A'k'y-  —  2nyy'-{-  Ay  '-  =  ;-. 

We  perceive  that  if  there  be  any  value  whatsoever  of  a  which  renders  n3B 
livisible  by  A,  this  value  can  be  augmented  or  diminished  by  any  multiple  of 
A,  without  ?r--B  ceasing  to  be  divisible  by  ce,  we  can  suppose  that 

its  value  is  comprised  between  the  limits  0  and  A,  or  even  between  the  more 
extended  limit  \  — .'A  and  +'  ^ • 

We  conclude  from  this,  that  in  trying  successively  for  n  all  the  entire  num 
bers  from  — 'A  to  -\-} ,A,  we  shall  encounter,  necessarily,  one  or  more  values 
which  will  render  n" — B  divisible  by  A,  provided,  howevor,  the  equation  is 
resolvable  ;  and  in  case  these  values  will  not  render  n- — B  divisible  by  A,  we 
can  conclude  with  certainty  that  the  proposed  equation  is  not  resolvable. 

441.  Suppose,  then,  that  we  have  found  one  or  more  values  of  ?i  which 
fulfill  the  required  condition,  tf  will  be  necessary  with  each  of  these  vain 
continue  the  calculation  in  the  following  manner  : 

Resume  the  equation  A'k-y:  —  2nyy'-\-Ay'*=:z*;  if  we  multiply  it  by  A'k* 
and  if  we  make,  to  abridge, 

A'k"y — ny'=x',  kz  =  z', 

tbe  transformed  will  be 

x'x'—By'y'=A'z,z' 

This  transformed  could  be  resolved,  if  we  could  determine  the  solution  of 
the  proposed  equation,  since  the  values  of  x',  y',  z'  are  easily  deduced  from 
those  of  x,  y,  z  ;  reciprocally,  the  proposed  will  be  resolved,  if  we  find  the  solu 
tion  of  ils  transformed.  For,  from  the  known  values  of  x',  y',  z',  we  cac 
equally  deduce  those  of  x,  y,  z  ;  and  it  matters  little  whether  these  last  value- 
be  under  an  entire  or  fractional  form,  since  we  have  regard  only  to  the  resolu 
tion  in  rational  numbers,  and  since,  after  we  have  found  any  fractional  values 
of  x,  y,  z,  we  can  reduce  them  to  a  common  denominator  and  suppress  it. 

Since  we  can  suppose  the  number  n  <.'  A,  it  is  clear  that  "Try-  or  A'  will 

be  <C}A,  and,  at  the  same  time,   positive;  for  n  can  not  be  <  VB,  since 

otherwise   n" — B   would   be    <B,  and  could  not  be   divisible  by   A.     The 

osed  equation,  therefore,  will  be  reduced  to  ;ion  in  every  respect 

similar,  in  which  the  coefficient  A',  which  takes  the  place  of  A,  is  less  than 

'.  If  we  have,  again,  A'>B,  wo  can,  in  like  manner,  from  the  equa 
'-• — By'"=zA'z':,  deduce  a  second  transformed, 

-n/'-=A' 

in  which  A."  will  be  <JA',  '""l  always  positive.    To  obtain  this  Becond  trans- 
formed, there  will  be  do  new  condition  to  bo  full'  [ready  fa 


GAUSS'S  METHOD  OF  SOLVING  BINOMIAL  EQUATIONS.  501 

■B 


=Aft3,  if  we  make  n=/iA'+n',  and  if  we  take  the  indeterminate  /*  in 
A 

n'2— B 
such  a  way  that  «/<^A',  it  is  easy  to  see  that  — -r-, —  will  be  an  entire  positive 

number  less  than  ]  A' ;  we  have,  consequently, 

n'2  — B  =  A'A"7c'3, 
A"  being  less  than  JA',  and  not  containing  any  square  factor. 

If  it  should  happen  that  A",  again,  were  greater  than  B,  we  should  continue 
this  system  of  transformed  equations,  in  which  B  is  constaut,  until  we  arrive 
at  one  of  this  form 

x2— By2  =  Cz\ 
in  Which  C  will  be  positive  and  <B. 

443.  But  after  we  have  passed  into  the  second  member  of  this  equation  the 
term  which  has  the  greatest  coefficient,  which  gives 

x2— Cz"-=By2, 
we  can  proceed  in  a  similar  maimer  to  the  roduction  of  the  coefficient  B  by  a 
second  system  of  transformed  equations 

x"2  —  Cz'2=B'y'2. 

x"2—Cz"2=B"y"2, 

&c.  &c, 

in  which  the  coefficients  B',  B",  &c,  will  be  positive,  and  will  diminish  in  at 
least  a  quadruple  ratio,  and  thus  we  shall  soon  arrive  at  a  transformed 

x2— Cz*=~Dy2, 
in  which  the  coefficient  D  will  be  less  than  C. 

But  the  series  of  positive  arid  decreasing  numbers  A,  B,  C,  D  will  not  go 
on  ad  infinitum ;  it  will  terminate  necessarily  at  unity,  and  when  we  shall 
have  arrived  at  this  term,  the  resolution  of  the  last  transformed,  which  is  given 
at  once,  will  make  known  tho?e  of  all  the  preceding  equations,  and,  consequent- 
ly, that  of  the  proposed. 

GAUSS'S  METHOD  OF  SOLVING  BINOMIAL  EQUATIONS. 

444.  The  solution  ofx"  —  1  =  0,  it  has  been  proved  (Art.  299),  can  al- 
ways be  reduced  to  tho  caso  where  n  is  prime;  ami  the  case  of  n  a  prime 
number,  by  a  method  invented  by  Gauss,  may  be  made  to  depend  upon  the 
solution  of  equations  whose  degrees  do  not  exceed  the  greatest  prime  number 
which  is  a  divisor  of  n  —  1.  The  leading  feature  of  Gauss's  method  is  to  rep- 
resent the  imaginary  roots  by  a  series  of  powers  of  any  one  of  them,  whose 
indices  form  a  geometrical  instead  of  an  arithmetical  progression.  Thus,  if  m 
be  a  number  (and  such,  called  primitive  roots  of  n,  can  always  be  found)  whose 
several  powers  from  1  to  n — 1,  when  divided  by  n,  leave  different  remainders, 
and  a  be  any  imaginary  root,  then  all  tho  roots  may  manifestly  be  represent- 
ed by 

om,  am",  a"'3,  .  .  .  a"1"-1  ; 

or,  since  mn_1  =tun -4-1,  where  /t  is  an  integer,  by  a,  am,  cm3,  &c,  a™"-*. 

445.  The  advantage  of  this  mode  of  representing  the  roots  is,  (1)  that  they 
can  bo  distributed  into  periods,  each  of  which,  when  continued,  will  produre 
the  roots  of  that  period  i'i  the  f  :ne  order;  and  (  ')  tha 


ALGEBRA, 
till 


r  of  such  periods  will  be  equal  to  the  sum  of  a  certain  number  of  periods, 
tho  importance  of  which  properties  will  be  seen  in  the  use  made  of  them. 

(1)  Let  n — l=rs,  r  being  a  prime  factor  of  n  —  1,  and  let  mT=.li ;  then  the 
roots  may  be  written  in  vertical  columns,  each  consisting  of  r  terms,  as  follows  : 


a  ah  a1'*  .  .  .  a'1' 

am  amh  amli  _  <  <  amli 


~r— 1  r— 1.  •     1,  2  r— 1,,b — 1 

am  am        b    am       ^     .  .  .  a">        b        , 


and  if  any  one  of  the  periods  formed  by  the  horizontal  rows  be  continued,  tne 
roots  in  that  period  will  be  produced  in  the  same  order;  thus,  if  the  first 
row  were  continued,  the  indices  would  be  /i'r=?»r3=mn_1=^«  +  l'  7i,+1=ttl^*",", 
=  (lin^-l)inr=imh-\-h,  &c,  and  the  corresponding  roots,  a,  a'1,  &c. 
(2)  Let  any  two  of  the  above  periods  be  represented  by 

a*  +olb  +  ^^4.,  &c,  4-aah5_1 
ab_^abh_|_abh2_|_?  &C-)  ^.abh8-1, 

and  let  us  multiply  them  together,  using  each  term  of  the  lower  line  in  suc- 
cession as  a  multiplier,  and  starting  at  that  term  of  tho  upper  line  which  stands 
over  it,  and  producing  the  upper  line  so  as  to  supply  the  terms  neglected  at 
the  beginning,  the  result  is 

a3+b  _^.a»H-b  -r-a;lh'+b  -f,  &c,  4-aab9-1+b 

a(a+b)l>        _|_a(ah+b)h        _|_a(nb2+b)h         _L_ ,    &C.,    +  a:^8_1-H>)h 
ata+bjb3       _J_a(ab+b)b2      4_a(ab2+b)b2       4.,    SCC,    -f  0  U,'S_'  +  b)h* 

a(a+b)b8-1^_a(ab+b)U,-1_^a(aU3+b)^-I_J_i    ^CC,    -f  C  ^-1+b)!»',~l  ; 

and  therefore,  collecting  the  vertical  columns  into  periods,  we  get 

2(aa)S(ab)  =  S(^+b)4.v(aai,+b)_|_v(ft,b-+b^#i-..   ,         ~1+^), 

or  the  product  of  two  periods  is  equal  to  the  sum  of  5  periods  ;  and,  conse- 
quently, the  product  of  any  number  of  periods  will  be  equal  to  the  aggregate 
of  a  certain  number  of  periods. 

kxample  1. 

x'  — 1=0  ;  6=3.2,  .-.  r=3,  s  =  2  ;  also,  3,  3:,  3\  3',  3\  when  divided  by  7, 
leave  different  remainders,  viz.,  3,  2,  G,  4,  5  ;  .-.  m  =  3,  and  the  roots  are 

pi  =  a  -\-aG 
/':=a?-\-a* 
/>—.;' 4- a6, 

and  pi4-pa+jp3=— 1- 

AlflC  plpi  =  a*  4-  .r-j-ir'  -\-<i- =,]>.  +  /,. 

]>:}).<  =  aP  -f  Cfi  4-  a  -|-  if-  =p  1  -f  /)., 
plp3  =  a3-\.a  4-a''4-„<=/>  4 

••■lh/>:-\  ]>■!>;  + I'll'.—  —  ■■'• 

nn-l  ^i^.;':1=/'.-  +  /''  +  /,-  =  ':!+/'-'+/'.+/,='1- 

Therefore  the  cubic  which  has  /-,.  y  .  ,\  For  it-  ro  >te,  is  /''+/'■— ~p  — l=i» 


GAUeS'S  METHOD  OE  SOLVING  BINOMIAL  EQUATIONS.  503 

EXAMPLK  II. 

c17  — 1  =  0  ;  1G=2.8 ;  also,  the  powers  of  3  from  0  to  15,  wheD  divided  by  17 
leave  remainders 

1  3  9  10  13  5  15  11  16  14  8  7  4  12  2  6, 
.-.  pssa  +a°  4.a13+a:8-i-a1«4-a8+a<  +  a2 
9=a3  4-a1"+ar'+a114-a1«-r.a74-a12+a«; 

tuen  p-\-q= — 1,  and 

p<7  =  a»-f  a12-f  a^-f  a  -f  a2-f  an-f  a7  -fa5 
a2-f  a"  _f  a8  4-o9+a  -f  au-f  a12-f  a" 

a8 -fa7  4-ai5_|.a2_j_a4_j_a5  _|_0u^_aio 

=P+1  +P  +P+P+Q  +7  +'?=— 4; 
therefore,  p  and  7  are  roots  of  z2-f  z — 4=0. 

Next,  the  periods  p,  q  may  bo  resolved  respectively  into  the  periods 
r  =  a  -f  a13-f  a,6-f  a4?       £  =  a3 -f  a5 -f  au-f  a12  > 
S  =  a9-f  a''-f  a8 -f  a2  V   w  =  a10-f  an-f  a7 -f  a^   >   ' 
.-.  r+s—p, 
and  rs = a10-f  a5  -f  a8  -f  a13 ' 

aii_|_a144-a2+alli 


^  =p+'?= — 1 ; 


as  -fa3  -f  a15-f  a 

therefore,  r,  s  are  roots  of  z- — pz  — 1  =  0  ;  and,  similarly,  t,  u  are  roots  of 
—  qz  — 1=0. 
Lastly,  the  periods  r,  s,  t,  u  may  be  resolved  respectively  into 


ri=a   +aIG>     sl  =  a^  -fa8  >     <1  =  a3-f  a14  )     tti=a10-fa7 
r,=a13-f  a'1   )  '  s2  =  a15-f  a2  S  '  ^  =  a5-f  a12  S  '  w2  =  au  +  a6 


then  ri-fr8=r, 

r1r2=a14-f  «H+ «3+ <zf>=<, 
.•.  r,,  r2  are  roots  of:2 — rz-f  £=0  ; 

Iff 

and  n,  the  greatest  root  of  this  equation,  =a-f  -=2  cos  — . 

For  further  information  upon  the  theory  of  numbers,the  student  is  referred 
to  the  TliCorie  des  Nombres  of  Legendro,  the  Disquisitiones  Arithmetics  of 
Gauss,  of  which  there  is  an  excellent  French  translation  (Recherches  Arith- 
metiqucs)  by  Poullet-Delisle ;  to  Barlow's  Theory  of  Numbers  ;  to  the  article 
of  Ivory  in  the  fourth  volume  of  the  supplement  of  the  Encyc.  Britan. ;  and 
to  the  Memoirs  by  Libri,  in  tome  v.,  1838  (JEtrangcres),  and  by  Cauchy,  iu 
tome  xvii.,  1840,  of  the  MCmoires  of  the  French  Academy  of  Sciences. 


S.  4,685  5713;  V.  — 0 

: 

- 

2  ; 

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loci.  36 2:. 

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